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Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006
VOLUME I Plenary Lectures and Ceremonies Marta Sanz-Solé Javier Soria Juan Luis Varona Joan Verdera Editors
S E M M uropean athematical ociety E S E M S ICM_vol1_titelei 26.3.2007 16:17 Uhr Seite 4
Editors:
Marta Sanz-Solé Juan Luis Varona Facultat de Matemàtiques Departamento de Matemáticas y Computación Universitat de Barcelona Universidad de La Rioja Gran Via 585 Edificio J. L. Vives 08007 Barcelona Calle Luis de Ulloa s/n Spain 26004 Logroño Spain Javier Soria Departament de Matemàtica Aplicada i Anàlisi Facultat de Matemàtiques Joan Verdera Universitat de Barcelona Departament de Matemàtiques Gran Via 585 Universitat Autònoma de Barcelona 08007 Barcelona 08193 Bellaterra (Barcelona) Spain Spain
2000 Mathematics Subject Classification: 00Bxx
ISBN 978-3-03719-022-7
Bibliographic information published by Die Deutsche Bibliothek
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©2007 European Mathematical Society
Contact address:
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Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printed in Germany
9 8 7 6 5 4 3 2 1 Preface
When we started planning the edition of the Proceedings of the International Congress of Mathematicians 2006 (ICM2006), we considered the possibility of publishing only an electronic version. However, it is pretty difficult to break traditions, particularly for an activity like the ICM with an existence of more than a hundred years. Thus, we finally decided to mimic the model that started in Berlin 98: to publish both, a printed and an electronic version of the Proceedings. However, you may notice the influence of living the Internet Era, where length of files is not really a big issue, by the number of pages, altogether almost 4400, probably a record for the history of ICMs. These Proceedings consist of three volumes. Volume I is divided into four parts. The first one gathers the speeches at the opening ceremony including the presentation of the Fields Medals, the Rolf Nevanlinna Prize and the newly awarded Gauss Prize for Applications of Mathematics as well as the speeches at the closing ceremony. It also contains information about the organization of the Congress, the committees, sponsors and other collaborators. The second part contains the traditional laudationes for the prizes, that is, an extensive presentation of the work of the awardees. The third part is the main body of the volume and consists of the articles written by the plenary lecturers of the Congress. One of the characteristics of this ICM has been the large number of diverse activities accompanying day by day the program fixed by the IMU Scientific Program Committee. In the fourth part of the volume, you can find articles corresponding to some of them. Volumes II and III were printed before the Congress and distributed to the partici- pants in Madrid. They gather the articles written by the invited speakers in the different scientific sections of the Congress. The on-line version of these volumes is accessible at the address http://www. icm2006.org/proceedings We take this opportunity to express our thanks to the authors of the articles for their effort in the preparation of excellent contributions. We also would like to express our gratitude to the EMS Publishing House for the superb job in the edition of these Proceedings and all the printed material of the ICM2006.
March 2007 Marta Sanz-Solé Javier Soria Juan Luis Varona Joan Verdera
Contents
Preface ...... v Past congresses ...... 1 Past Fields Medalists and Rolf Nevanlinna Prize Winners ...... 2 Organization of the Congress ...... 3 The committees of the Congress ...... 13 Other collaborators of the ICM2006 ...... 22 List of sponsors ...... 23 Opening ceremony ...... 25 Closing ceremony ...... 45
The work of the Fields Medalists, the Rolf Nevanlinna Prize Winner and the Gauss Prize Winner ...... 53 Giovanni Felder The work of Andrei Okounkov ...... 55 John Lott The work of Grigory Perelman ...... 66 Charles Fefferman The work of Terence Tao ...... 78 Charles M. Newman The work of Wendelin Werner ...... 88 John Hopcroft The work of Jon Kleinberg ...... 97 Hans Föllmer On Kiyosi Itô’s work and its impact ...... 109
Plenary Lectures Percy Deift Universality for mathematical and physical systems ...... 125 Jean-Pierre Demailly Kähler manifolds and transcendental techniques in algebraic geometry .... 153 Ronald A. DeVore Optimal computation ...... 187 Yakov Eliashberg Symplectic field theory and its applications ...... 217 viii Contents
Étienne Ghys Knots and dynamics ...... 247 Henryk Iwaniec Prime numbers and L-functions ...... 279 Iain M. Johnstone High dimensional statistical inference and random matrices ...... 307 Kazuya Kato Iwasawa theory and generalizations ...... 335 Robert V. Kohn Energy-driven pattern formation ...... 359 Ib Madsen Moduli spaces from a topological viewpoint ...... 385 Arkadi Nemirovski Advances in convex optimization: conic programming ...... 413 Sorin Popa Deformation and rigidity for group actions and von Neumann algebras .... 445 Alfio Quarteroni Cardiovascular mathematics ...... 479 Oded Schramm Conformally invariant scaling limits: an overview and a collection of problems ...... 513 Richard P. Stanley Increasing and decreasing subsequences and their variants ...... 545 Terence Tao The dichotomy between structure and randomness, arithmetic progressions, and the primes ...... 581 Juan Luis Vázquez Perspectives in nonlinear diffusion: between analysis, physics and geometry ...... 609 Michèle Vergne Applications of equivariant cohomology ...... 635 Avi Wigderson P , NP and mathematics – a computational complexity perspective ...... 665
Special activities John W. Morgan The Poincaré Conjecture ...... 713 Panel discussion organised by the European Mathematical Society Should mathematicians care about communicating to broad audiences? .... 737 Contents ix
ICM 2006 Closing round table Are pure and applied mathematics drifting apart? ...... 757
Cultural activities José M. Sánchez-Ron The road from Zurich (1897) to Madrid (2006) ...... 777
List of participants ...... 795 Participants by country ...... 830 Author index ...... 831
Past congresses
1897 Zurich 1958 Edinburgh 1900 Paris 1962 Stockholm 1904 Heidelberg 1966 Moscow 1908 Rome 1970 Nice 1912 Cambridge, UK 1974 Vancouver 1920 Strasbourg 1978 Helsinki 1924 Toronto 1982 Warsaw (held in 1983) 1928 Bologna 1986 Berkeley 1932 Zurich 1990 Kyoto 1936 Oslo 1994 Zurich 1950 Cambridge, USA 1998 Berlin 1954 Amsterdam 2002 Beijing
Madrid 2006 2 Past Fields Medalists and Rolf Nevanlinna Prize Winners
Fields Medalists
1936 Lars V. Ahlfors 1978 Pierre R. Deligne Jesse Douglas Charles F. Fefferman Grigorii A. Margulis 1950 Laurent Schwartz Daniel G. Quillen Atle Selberg 1982 Alain Connes 1954 Kunihiko Kodaira William P. Thurston Jean-Pierre Serre Shing-Tung Yau
1958 Klaus F. Roth 1986 Simon K. Donaldson Rene Thom Gerd Faltings Michael H. Freedman 1962 Lars Hörmander John W. Milnor 1990 Vladimir G. Drinfeld Vaughan F. R. Jones 1966 Michael F. Atiyah Shigefumi Mori Paul J. Cohen Edward Witten Alexander Grothendieck Steve Smale 1994 Jean Bourgain Pierre-Louis Lions 1970 Alan Baker Jean-Christophe Yoccoz Heisuke Hironaka Sergei P. Novikov 1998 Richard E. Borcherds John G. Thompson William T. Gowers Maxim Kontsevich 1974 Enrico Bombieri Curtis T. McMullen David B. Mumford 2002 Laurent Lafforgue Vladimir Voevodsky
Rolf Nevanlinna Prize Winners
1982 Robert E. Tarjan 1994 Avi Wigderson 1986 Leslie G. Valiant 1998 Peter W. Shor 1990 Alexander A. Razborov 2002 Madhu Sudan Organization of the Congress
Manuel de León, President of the ICM2006
In 1998, the Real Sociedad Matemática Española, the Societat Catalana de Matemà- tiques, the Sociedad Española de Matemática Aplicada and the Sociedad de Estadís- tica e Investigación Operativa got together to reorganize the Spanish Committee of Mathematics (CEMAT) representing Spain at the IMU. This Committee, which in- cludes three other societies (the Federación Española de Sociedades de Profesores de Matemáticas, the Sociedad Española de Investigación en Educación Matemática, and the Sociedad Española de Historia de las Ciencias y de las Técnicas), put forward the Spanish candidacy to host the 25th International Congress of Mathematics in Madrid in 2006, as well as the IMU General Assembly in Santiago de Compostela. This bid was initially backed by the City of Madrid, the Autonomous Community of Madrid, the Ministry of Education, Culture and Sport, the Ministry of Science and Technology and the Ministry for Foreign Affairs. His Majesty King Juan Carlos I also gave his support to the candidacy with a letter included in the dossier. In addi- tion, backing was forthcoming from the universities in the region (the Universidad Complutense de Madrid, the Universidad Autónoma de Madrid, and the Universidad Carlos III de Madrid) with letters from their respective rectors, as well as from the president of the Consejo Superior de Investigaciones Científicas (CSIC). An associa- tion was created to promote the candidacy, which brought together the support of all the above-mentioned bodies and institutions. The candidacy was advocated by the Spanish delegation headed by José Luis Fernández at the 24th General Assembly in Shanghai, and was unanimously approved by vote. The invitation to come to Madrid was formally made on behalf of Spain by Carles Casacuberta at the ICM2002 closing ceremony in Beijing. The association formed to present the candidacy was dissolved on its return from China, and work began on the organization of the ICM2006 in Madrid and the General Assembly in Santiago. To this end, the ICM2006 Madrid Association was set up, independently of the CEMAT and the societies but in complete co-ordination with all of them. The first president of this association was Carlos Andradas, who was replaced in 2003 by Manuel de León. At the same time, an Organizing Committee responsible for the General Assembly was set up at the Universidad de Santiago. This Committee included the three universities in the region (Santiago de Compostela, La Coruña and Vigo), and was headed by the dean of the Faculty of Mathematics, Juan Manuel Viaño. Both bodies have worked in full co-ordination with each other in recent years. A further important point is that, although the ICM2006 was to be held in Madrid, the organization of the congress was a joint effort across the whole country. In addition to the GeneralAssembly in Santiago, the Committee was composed of mathematicians from all over Spain, a reflection of the country’s historical and cultural wealth and variety. A consultation of the web page will reveal messages of welcome not only in 4 Organization of the Congress
English, but also Spanish, Catalan, Euskera and Galician; in other words, in all the official languages of the Spanish state. A major congress with a scope such as that of the ICM also requires strong financial and logistic support from public administration bodies, and as such is subject to political changes. This is precisely what occurred in the city and region of Madrid. The change of government in Spain in 2004 brought about a restructuring of ministries, and with it a corresponding change in our interlocutors, who became the Ministry of Education and Science, the Ministry of Culture, and the Ministry of Foreign Affairs and Cooperation. We are bound to state that the support shown by the previous government for the organization of the ICM was taken up by its successor, both of whom were fully aware of the unique importance of the event. With regard to financial support, the ICM2006 Executive Committee worked ex- tremely hard to achieve the following goals: 1) To secure the backing for the event from institutions; firstly the Committee of Honour was proud to have His Majesty the King as its president, with representatives of all the public authorities: the Prime Minister, other ministers, the mayor of Madrid, the president of the Regional Govern- ment, and the rectors and president of the CSIC; 2) To ensure solid public funding, which came from the Ministry of Education and Science, the Community of Madrid, Madrid City Hall and from the CSIC, and 3) To attract funding from the private sector, which eventually fell short of initial expectations, and which except for organizations such as the Vodafone Foundation, BSCH, the Areces and Enterasys Foundations, as well as Spanish companies and those with their headquarters in Spain, are still a long way from recognizing mathematics as a driving force in research, technological de- velopment and innovation. We Spanish mathematicians have also learned that this section on the road to understanding still remains to be covered. The organization of an ICM requires an important logistical underpinning that cannot be left to voluntary contributions. For that reason we chose a congress agency with great experience in organizing major events, and one with enough flexibility to adapt to our needs. This agency was Unicongress. With their team headed by Paloma Herrero we worked hand in hand as though the ICM2006 were indeed a joint venture, and together we shared the achievements and setbacks which, like all those who have been involved in previous ICMs, we know are part of and parcel of this difficult task. I am happy to say that our choice was the right one, and that the outcome was satisfactory for all concerned. We also believe that for Unicongress, too, this was a new experience, since any ICM amounts to much more than a conventional congress. The ICM congress logo is something that remains in the mind for years to come. It is not easy to devise a logo that embodies at once the essence of a country and mathematics itself. After many attempts we settled on the one that has since become familiar, and is inspired in the sunflower. One the one hand, the sunflower symbolizes the Spain of sun and light already known to many; on the other, the number of its spirals to right and left are elements of the Fibonacci sequence. The artistic creativity of its devisers led to an image that resembles both a sunflower and the fractal nature of a Romanesco cauliflower. This has given rise to different mathematical interpretations, Organization of the Congress 5 and even to different reproductions of the original. The colours of the logo can be fully appreciated on the congress website, where in the logo structure the different themes are associated with the different colours. The logo formed the basis for one of the official posters designed to promote the ICM2006, together with four others based on well-known pieces of Spanish archi- tecture with a mathematical content. These five posters were sent to mathematical departments the world over and have been very well received internationally. The target for the ICM2006 budget was 2 500 000 euros, including the attendance fees and specific help from the IMU. The fees could not be set too high, otherwise it would have prevented mathematicians from countries with economic difficulties from attending the congress. Thus following the custom of previous ICMs, it was set at 260 euros, which scarcely covered the expenses generated by each participant in terms of proceedings, coffee, congress bag, materials, etc… As mentioned before, most of the budget was provided by public sources and the fees. On-going work with the General Secretariat of Scientific and Technological Policy enabled us to meet all budget requirements without any final deficit. I would like to mention the outstanding work carried out by the Treasurer, Alberto Ibort, and the vice-Treasurer, MiguelÁngel Rodríguez, thanks to whom the accounts for the ICM2006 remained always on an even keel. Every ICM is special in some respect, and ours was no exception. The Committee wanted to emphasize three main branches or axes peculiar to the geo-strategic situation of Spain in history and in the world, in particular in relation to Europe. • The European axis, as a reflection of Spain’s position in Europe, symbolized by holding the General Assembly in Santiago de Compostela, the destination of pilgrims along the Road to Santiago, which acted as a channel for culture and science in the Middle Ages. • The Latin American axis, highlighting the existence of a cultural community by means of which Spain wishes to further its links, including those concerned with mathematics; and • The Mediterranean axis, with Spain as a bridge between Africa, the Near East and Europe, with the intention of increasing mathematical co-operation in this sphere. The venue for the congress deserves a section on its own. For a congress such as the ICM, eminently scientific in character, but also with its relevance in social and media terms, with the presentation of the most prestigious prizes awarded in mathematics, an appropriate venue is a crucial factor. The Palacio Municipal de Congresos (PMC) in Madrid is a striking building designed by Ricardo Bofill, one of the most highly recognized Spanish architects on the international scene. This majestic building, equipped with all the latest modern technology, provided everything we could have wished for. But this did not come cheaply, and in fact accounted for a considerable part of the budget. However, thanks to our collaboration with the Convention Bureau 6 Organization of the Congress of Madrid, Madrid City Hall, and those in charge at the PMC itself, we were able to secure the building as the venue for the congress. In retrospect, I believe our decision to have been the correct one, and it is true to say that the ICM2006 would not have been the same without these premises. The opening ceremony is another vital part of any ICM. For several months, we debated with the IMU Executive Committee, and in particular with its president, Sir John Ball, about how the ceremony would be structured. The presence of His Majesty the King at this opening ceremony on August 22nd was decisive for attaining the impact desired, and we are grateful for the extraordinary co-operation extended by the Royal Household from the very beginning. In spite of difficulties with the agenda, to say nothing of the security measures required, everything was in place on time for the event. Not only is financial support from public institutions necessary for a congress of this nature, but also the physical presence of their representatives. In this case, we the organizers would like to express our thanks to the Royal Household, to Madrid City Hall, to the Community of Madrid and the Ministry of Education and Science for all their support in both these respects. The opening ceremony was divided into two parts; the first part consisted of a video produced by the organizers showing the relation of mathematics with art and culture through the ICM2006 official posters. There was also a musical performance by the Ara Malikian Trio that enjoyed great success. The second part consisted of the official speeches and the presentation of the prizes by His Majesty the King. We believe this was an emotive and attractive event befitting the importance of the awards and the prize-winners themselves. Finally, His Majesty the King delivered a speech pointing out the vital role played by mathematics in education, knowledge and development. After his address, the King declared the Madrid ICM2006 officially open. His Majesty also attended the cocktail reception held after the opening ceremony, and delighted everyone with his cordiality and friendly approachability. After the opening ceremony, the congress unfolded according to plan. The quality of the lectures was a concern of both the Programme Committee and the Local Pro- gramme Committee, not only for their content, which was beyond all doubt, but also for the presentations. Noga Alon’s work on behalf of the PC, and Marta Sanz-Solé’s on behalf of the LPC, were both admirable, and I am sure I am not mistaken when I say that the ICM2006 fully emulated previous ICMs in this respect. There is no doubt that the technological facilities at the Palacio Municipal de Congresos did much to ensure the quality of both invited talks and plenary lectures. The scientific programme consisted of 20 plenary lectures and 169 invited talks distributed over 20 sections, the same amount as at the ICM2002. With regard to the open programme, the presentation of posters was encouraged by a competition with prizes for the best entries, a measure whose purpose was to make the programme more agreeable and digestible. Steps are taken at every ICM to encourage the participation of mathematicians from the more disadvantaged countries. Indeed, co-operation in development is a priority of the IMU, as explicitly stated in the resolutions approved at the 25th General Organization of the Congress 7
Assembly. On this occasion, the IMU and the ICM2006 established the following five categories for financial support: 1. Youngmathematicians from developing and economically disadvantaged coun- tries. 2. Senior mathematicians from developing and economically disadvantaged coun- tries. 3. Senior mathematicians from Latin America. 4. Senior mathematicians from Mediterranean developing countries. 5. Young Spanish mathematicians. The IMU subsidized the travel expenses of 143 mathematicians selected for Pro- grammes 1 and 2: 80 on Programme 1 and a further 63 on Programme 2, while the Local Organizing Committee covered the registration fee, board and lodging in Madrid for 131 of these 143 participants. In accordance with the three axes previously described, the ICM2006 Organizing Committee also managed to include Programmes 3, 4 and 5, covering the registra- tion fee, board and lodging in Madrid for 178 mathematicians (Programme 3: 76; Programme 4: 70; Programme 5: 32) and 43 airline tickets (Programme 3: 25; Pro- gramme 4: 18). These five programmes were co-ordinated by C. Herbert Clemens, Linda Geraci and Sharon Laurenti (IMU), and also by Marisa Fernández (ICM2006). This task was possible thanks to their efforts and dedication. The specific funding was provided by the IMU Special Development Fund, the Spanish Agency for International Cooperation (Ministry of Foreign Affairs and Coop- eration), the Departments of Mathematics and Deans of the Faculties of Mathematics of the Spanish Universities, and the Carolina Foundation, as well as by a large number of Spanish and non-Spanish mathematical societies. As regards co-operation, one of the activities undertaken prior to the ICM2006 was the “Mathematics for Peace and Development” School. During the week July 17th-23rd, young mathematicians from Arab countries (including Palestine), Latin America, Europe and Israel attended eight courses given at the Universidad de Córdoba by prestigious mathematicians from different countries. The aim of the School was to draw attention to mathematics as an effective means of contributing to the progress of peoples, as well as its use as a universal language for mutual understanding among different cultures. The choice of Córdoba as the venue was the role of this city as a symbol of the “Spain of the Three Cultures”, where Christians, Jews and Muslims lived side by side in an example of tolerance and co-operation. The Madrid ICM2006 was also complemented by 64 satellite conferences – a record. 36 of them were held in Spain and constituted a demonstration of the orga- nizational powers of Spanish mathematicians and their many international relations. There was no specialized branch of mathematics that was not addressed in any of the satellite conferences. Although at times these satellite conferences can draw 8 Organization of the Congress attendance away from the ICM itself, in this case the number and quality of such con- ferences more than made up for any shortfall and proved to be an excellent scientific accompaniment. Cultural and dissemination activities were other facets of the ICM that were ac- corded fundamental importance. In consequence, an ambitious programme was drawn up to cover two fronts: on the one hand, society in general, and on the other the congress participants. Our aim was to draw attention to the role played by mathemat- ics throughout the length and breadth of geography and history and in the culture of humankind, as well as showing how mathematics is an essential part of life. Judging by the results, and by the reactions to these cultural activities, which were praised in King Juan Carlos’ opening speech and in the closing speech by the IMU president, they were one of the outstanding successes of the congress. The responsibility for this task fell to a team led by Antonio J. Durán, in collaboration with Raúl Ibáñez, Guillermo Curbera and Antonio Pérez-Sanz. In relation with the ICM2006 in particular, and in the effort to bring mathematics closer to society at large, the exhibition “The Life of Numbers” was expressly prepared for the occasion, and was organized and financed by the Ministry of Culture and the Spanish National Library. The exhibition was held in Madrid at the Spanish National Library from June 7th to September 10th, and provided an account of the relation of human beings with numbers from the first marks left by human hands in Palaeolithic cave paintings to the Renaissance, a journey through Mesopotamia, Egypt, Greece, Mesoamerica, Rome, India and the Middle Ages. On display at the exhibition were Babylonian tablets, Roman coins, pre-Roman and Mayan manuscripts, an impres- sive collection of Renaissance mercantile arithmetics, engravings by Leonardo da Vinci and Durer, maps of the Earth and the Stars, all the exhibits coming from dif- ferent Spanish institutions: the Museum of America and the National Archaeological Museum, the Library of the Monasterio de El Escorial, the Capitular & Colombina Library in Sevilla, the Universidad Complutense de Madrid Library, from Catalo- nia, and of course from the Spanish National Library itself. The pièce de résistance was the Codex Vigilanus, a manuscript composed in 976 at the Monasterio de San Martín de Albelda (La Rioja), currently conserved at the Monasterio de El Escorial. This manuscript is the oldest written record of its kind in history and includes the Hindu-Arabic numerals which are still the basis of our numbers today. A beautifully illustrated edition of the book “The Life of Numbers” was published for the exhibition with texts by Alberto Manguel, Georges Ifrah and Antonio J. Durán (who was also the curator of the exhibition). Also with the general public in mind, three exhibitions were organized at the Centro Cultural Conde Duque in Madrid, financed by the Ministry of Education and Science and the Spanish Foundation for Science and Technology. Firstly, the already well-known “Experiencing Mathematics”, an exhibition originating at the French Centre des Sciences in Orleáns and sponsored by UNESCO. This exhibition was presented under the Spanish title of “¿Por qué las Matemáticas?” (“Why Mathe- matics?”) and was open at the Conde Duque Cultural Centre from August 17th to Organization of the Congress 9
October 20th (the curators being Raúl Ibáñez and Antonio Pérez Sanz). The sec- ond exhibition, organized expressly for the occasion, concerned fractal art and went under the title of “Fractal art: Beauty and Mathematics”. On display were works by the twenty-eight finalists in an international competition expressly organized for the ICM2006 in Madrid, with a jury of panellists headed by Benoit Mandelbrot. Professor Mandelbrot gave a talk on “The Nature of Roughness in Mathematics, Science, and Art” at the main congress venue, where a replicated version of this exhibition could also be seen. Highly visual catalogues were published for both exhi- bitions (the first included a notebook of activities for students). The third exhibition was “Demoscene: Mathematics in Movement”, held in parallel at the Centro Cultural Conde Duque and the congress venue, and consisted of a selection of computer- aided animated films with live commentary by some of their creators. Desmocene is a powerful source of mathematical algorithms for the creation of graphic and visual effects, whose special digital effects are currently used in feature films and video games. The success of all these exhibitions, whose purpose was to stimulate interest about mathematics in society at large, together with the celebration of the ICM2006 in itself, can be measured by their repercussion in the media and by the large number of people who came to visit them, to the extent that they frequently had to queue to enter. Those in charge at the Spanish National Library and the Centro Cultural Conde Duque were frankly surprised by the number of visitors, given the subject-matter of the respective exhibitions. The ICM2006 Executive Committee also mounted an extensive programme of activities for the Congress participants themselves. The most ambitious of these events was the exhibition entitled “The ICM through History”, based on the history of the 25 ICMs held to date, from the first held in Zurich in 1897 to the Madrid congress in 2006. The aim of the exhibition was to provide a visual chronicle of all the ICMs, emphasizing their significance in terms of human endeavour and using the activities of mathematicians at the ICMs as a mirror in which history, culture, technology, fashion and changing attitudes were reflected. Some 500 written and photographic documents provided a twin portrait of the ICMs; on the one hand, a chronological review of the history of the ICM, and on the other a transversal view through the social life of the congresses, the graphic design for the congresses and the buildings where they have been held. The physical and conceptual heart of the exhibition resided in the display of medals, original reproductions of the Fields, Nevanlinna and Gauss awards provided by the Royal Canadian Mint, the University of Helsinki and the Deutsche Mathematiker- Vereinigung. Guillermo Curbera, the curator of the exhibition, was helped in his task by many universities, libraries, archives, museums, mathematical societies and individuals, enabling him to assemble an extraordinary collection of photographs and documents, many of them never available to the public before. The exhibition was entirely financed by the ICM2006 Executive Committee and has remained as an asset of the Spanish mathematical societies. 10 Organization of the Congress
Another cultural activity that aroused much public and media attention was the Japanese sculptor Keizo Ushio’s live sculpting of a square block of black granite weighing various tonnes. From this he fashioned a torus which he split into two curved sections to form a sculpture resembling the symbol for infinity. Ushio began work in early August on the campus of the Consejo Superior de Investigaciones Científicas, from where he moved to the Congress venue on August 22nd. It was there he completed the work in full view of congress participants and passers-by, producing a sculpture that has attracted much attention, especially in Spain and Japan. This programme of activities was complemented by others which, although not organized directly by the Committee, were included in the general programme. One of the most noteworthy was the exhibition based on classical mathematical texts under the title of the “History of Mathematical Knowledge”, which was held at the “Marqués de Valdecilla” Historical Library of the Universidad Complutense de Madrid between June 28th and October 27th (with Ricardo Moreno as curator). Another was the replicated version of the exhibition organized by the University of Vienna, with Karl Sigmund and John Dawson as curators. This exhibition commemorated the centenary of Kurt Gödel and took place at the Botanical Garden of the Universidad Complutense de Madrid fromAugust 22nd to September 8th (with Capi Corrales Rodrigáñez as local co-ordinator). Further exhibitions were: “Singularities”, mounted at the Congress venue by professor Herwig Hauser (including a film show); a tribute to the musician Francisco Guerrero (including a concert held at the venue), and the mathematical visit to the Monasterio de El Escorial and its library (tickets for this event were sold out six months before the Congress began). Pride of place among the cultural events was the official gift presented to all plenary lecturers and invited speakers by the Executive Committee in recognition for their contribution to the Congress. This consisted of a facsimile edition of the works of Archimedes, “On the Sphere and the Cylinder”, “On the Measurement of the Circle” and “The Quadrature of the Parabola” (published jointly with the Real Sociedad Matemática Española), in an annotated Spanish translation. This is a luxury edition comprising two volumes presented in a box-set (333 × 230 mm). The first volume is a facsimile book of a 16th century manuscript from the Library of El Monasterio de El Escorial, a manuscript copied in Venice at the expense of Diego Hurtado de Mendoza (Charles V’s ambassador at Venice from 1527 to 1547) from the manuscript CCCV extant in the Marciana Library. The second volume contains the annotated Spanish translation of those Archimedean works, and the following four studies: (1) Greek Science: Towards a Critical Knowledge by Carlos García Gual; (2) Archimedes and His Manuscripts by Antonio J. Durán (who was also in charge of co-ordination of the edition); (3) Archimedes: A Legend of Wisdom, and (4) The Mathematical Works of Archimedes, both by Pedro M. González Urbaneja. Many other special activities were organized, a list of which would be too long to include in this introduction, although we may mention the scientific part of the Emmy Noether Talk, given by Ivonne Choquet-Bruhat, the special talk on Poincaré’s Conjecture by John Morgan, and the talk given by Benoit Mandelbrot. A joint scien- Organization of the Congress 11 tific activity organized by the London Mathematical Society and the Real Sociedad Matemática Española was also held. Several round table discussions were also held, among which were those organized by the European Mathematical Society onAugust 23rd, “Should Mathematicians Care about Communicating to Broad Audiences? Theory and Practice”, chaired by Jean Pierre Bourguignon and with the participation of Björn Engquist, Marcus du Sautoy, Alexei Sossinsky, François Tisseyre and Philippe Tondeur, and the ICM2006 Closing Round Table on August 29th, “Are Pure and Applied Mathematics Drifting Apart?”, chaired by John Ball and with the participation of Lennart Carleson, Ronald Coifman, Yuri Manin, Helmut Neunzert and Peter Sarnak. One of the most long-standing traditions in the history of the ICM is the edition of a special commemorative stamp. On this occasion, the design for the stamp included the congress logo and the first known written record of the Hindu-Arabic numbers from the Codex Vigilanus, published in Spain in the 10th century, which is currently conserved at the Library of the Monasterio de El Escorial on the outskirts of Madrid. The volunteers are a collective who deserve a special mention. We also wanted this group to be composed of representatives from all over Spain, and indeed volunteers came forward from all the Spanish universities. Some 700 pre-graduate and pre- doctoral grant students responded to our call, from which a total of 350 were selected. These volunteers worked hard and enjoyed the experience to such an extent that they were sad to see the ICM2006 come to a close, which in itself stands as a testimony to the success of their efforts. We on the Organizing Committee are indebted to all of them. They worked tirelessly for long hours without complaint, and I hope that many of them will be able to participate in the ICM2010 as fully-fledged mathematicians. Every ICM at its conclusion is obliged to present statistics providing an account in numbers of all that took place. The final figure of participants reached 3,600, with 400 accompanying persons. The number of countries from which participants came set an all-time record of 108. The number of exhibitors rose to 45. In the scientific part of the congress there were 20 plenary lectures, 169 invited talks and some 1,000 short communications and posters. There is no doubt that the most outstanding feature of this congress was the ex- traordinary attention it received from the media. In this regard, some have attributed this interest to the conspicuous absence of Grigory Perelman and his refusal to receive the Fields Medal, but it must be said that one year before the start of the congress the Organizing Committee set up a press office with the “Divulga” agency. Over the 20 weeks immediately prior to August 22nd, a weekly news bulletin providing informa- tion about the contents of the coming ICM was published. At the same time, the ICM public presentations and the most important parallel activities were programmed. Our press team headed by Ignacio F. Bayo and Mónica Salomone also collaborated with the IMU Executive Committee at an international level. Indeed, we sent letters to the leading communications media in Spain and abroad inviting their representatives to the opening ceremony. The combination of all these circumstances made the event a media success. For ten days during the summer in Spain the ICM2006 was headline 12 Organization of the Congress news, and international repercussion was likewise unprecedented. The lesson to be learned from this is that we mathematicians must work hand in hand with journalists and the media if we wish to emerge from the information ghetto. Press work culminated in the publication of 7 issues of the “Daily News”, often produced against the clock and in spite of permanent pressure on the press office arising from the continuous avalanche of requests for information from the media and its representatives. In addition, press conferences were organized on a daily basis, which sometimes attracted audiences hitherto unconceivable in the world of mathematics. With regard to diffusion, it is also necessary to mention the series of programmes produced by the UNED Educational Television and provided for the Organizing Com- mittee. These programmes constitute documents of great educational value. The Closing Ceremony was held on August 30th and featured the expressions of gratitude and acknowledgement from the IMU president to the different committees, my own to the committees who worked on the organization in Spain, the address by the elected president of the IMU, László Lovász, and the invitation from the Indian representative, Rajat Tandon, to attend the ICM2010 to be held in his country, in the city of Hyderabad. After every ICM there still remains work to be done. In addition to the thick VolumesII and III forming part of the Proceedings, there is the first that the reader now holds in his or her hands. The Publishing House of the European Mathematical Society was charged with the publication of these proceedings, and the result has certainly been impressive. This first volume is accompanied by a DVD with recordings of the opening and closing ceremonies, as well as all the plenary lectures. We believe that it provides an excellent complement to the text and an unforgettable record for all those who shared with us those wonderful ten days in Madrid in August 2006. This ICM2006 will have a long-lasting effect on Spanish mathematics. It has been a collective effort that has brought us closer together and made us aware of belonging to a national and international community. On a domestic level, it has also led to a self-examination that has given rise to initiatives that are already under way to improve research in the discipline. Furthermore, it has brought mathematics more to the social forefront to an extent never before witnessed in Spain. This is a situation that we must make the most of in the years to come. Moreover, the eyes of the international mathematical collective were fixed on Spain this summer, a fact that will undoubtedly further greater collaboration. We hope to have fulfilled all the expectations placed in us by the IMU, and leave the mathematical doors of our country open to the future. The committees of the Congress
Organizing committees
Honorary Committee President His Majesty, The King of Spain Members The Prime Minister of Spain The President of the Community of Madrid The Minister of Education and Science The Minister of Culture The Minister of Foreign Affairs The Minister of Industry, Tourism and Trade The Mayor of the City of Madrid The Rector of the Universidad Complutense de Madrid The Rector of the Universidad Autónoma de Madrid The Rector of the Universidad Politécnica de Madrid The Rector of the Universidad de Alcalá de Henares The Rector of the Universidad Carlos III de Madrid The Rector of the Universidad Rey Juan Carlos The Rector of the Universidad Nacional de Educación a Distancia The President of the Consejo Superior de Investigaciones Científicas
Executive Committee President Manuel de León, Instituto de Matemáticas y Física Fundamental, CSIC, Madrid Vice President General Carlos Andradas, Universidad Complutense de Madrid Vice Presidents Carles Casacuberta, Universitat de Barcelona Eduardo Casas, Universidad de Cantabria, Santander Pedro Gil Álvarez, Universidad de Oviedo Secretary General José Luis González-Llavona, Universidad Complutense de Madrid Treasurer Alberto Ibort Latre, Universidad Carlos III de Madrid
Cultural Activities Antonio J. Durán, Universidad de Sevilla 14 The committees of the Congress
Fund Raising & Sponsorship Emilio Bujalance, Universidad Nacional de Educación a Distancia, Madrid María Luisa Fernández, Euskal Herriko Unibertsitatea, Bilbao Infrastructure and Logistics Emilio Bujalance, Universidad Nacional de Educación a Distancia, Madrid Local Program Committee Marta Sanz-Solé, Universitat de Barcelona Parallel Scientific Activities Fernando Soria, Universidad Autónoma de Madrid Publications Joan Verdera, Universitat Autònoma de Barcelona Relations with Latin America, Eastern Europe and Developing Countries María Luisa Fernández, Euskal Herriko Unibertsitatea, Bilbao Social Activities Rosa Echevarría, Universidad de Sevilla Vice Treasurer Miguel Ángel Rodríguez, Universidad Complutense de Madrid Web and Electronic Communications Pablo Pedregal, Universidad de Castilla-La Mancha, Ciudad Real
Cultural Activities Chair Antonio J. Durán, Universidad de Sevilla Members Antonio F. Costa, Universidad Nacional de Educación a Distancia, Madrid Guillermo P. Curbera, Universidad de Sevilla Raúl Ibáñez, Euskal Herriko Unibertsitatea, Bilbao Infrastructure and Logistics Chair Emilio Bujalance, Universidad Nacional de Educación a Distancia, Madrid Members Roberto Canogar McKenzie, Universidad Nacional de Educación a Distancia, Madrid Francisco Javier Cirre Torres, Universidad Nacional de Educación a Distancia, Madrid Miguel Delgado Pineda, Universidad Nacional de Educación a Distancia, Madrid M. José Muñoz Bouzo, Universidad Nacional de Educación a Distancia, Madrid Ana M. Porto F. Silva, Universidad Nacional de Educación a Distancia, Madrid The committees of the Congress 15
Local Program Committee Chair Marta Sanz-Solé, Universitat de Barcelona Members Jesús Bastero, Universidad de Zaragoza José A. Carrillo, ICREA and Universitat Autònoma de Barcelona Wenceslao González-Manteiga, Universidade de Santiago de Compostela Consuelo Martínez, Universidad de Oviedo Marcel Nicolau, Universitat Autònoma de Barcelona Tomás Recio, Universidad de Cantabria, Santander J. Rafael Sendra, Universidad de Alcalá de Henares Juan M. Viaño, Universidade de Santiago de Compostela
Parallel Scientific Activities Chair Fernando Soria, Universidad Autónoma de Madrid Members Manuel Barros, Universidad de Granada Miguel Escobedo, Euskal Herriko Unibertsitatea, Bilbao Ignacio García Jurado, Universidade de Santiago de Compostela Luis Narváez Macarro, Universidad de Sevilla
Publications Editors of the Proceedings Marta Sanz-Solé, Universitat de Barcelona Javier Soria, Universitat de Barcelona Juan Luis Varona, Universidad de La Rioja Joan Verdera, Universitat Autònoma de Barcelona The Madrid Intelligencer Fernando Chamizo, Universidad Autónoma de Madrid Adolfo Quirós, Universidad Autónoma de Madrid
Relations with Latin America, Eastern Europe and Developing Countries Chair María Luisa Fernández, Euskal Herriko Unibertsitatea, Bilbao Members Antonio Cuevas, Universidad Autónoma de Madrid Eugenio Hernández, Universidad Autónoma de Madrid Ignacio Luengo, Universidad Complutense de Madrid 16 The committees of the Congress
Marta Macho, Euskal Herriko Unibertsitatea, Bilbao Raquel Mallavibarrena, Universidad Complutense de Madrid José Leandro de María, Universidad Nacional de Educación a Distancia, Madrid Ernesto Martínez, Universidad Nacional de Educación a Distancia, Madrid Vicente Muñoz, Instituto de Matemáticas y Física Fundamental, CSIC, Madrid Domingo Pestaña, Universidad Carlos III de Madrid José Manuel Rodríguez, Universidad Carlos III de Madrid
Web and Electronic Communications Chair Pablo Pedregal, Universidad de Castilla-La Mancha, Ciudad Real Member Ernesto Aranda Ortega, Universidad de Castilla-La Mancha, Ciudad Real
IMU committees
Program Committee Chair Noga Alon, Tel Aviv University, Israel Members Douglas N. Arnold, University of Minnesota, USA Joaquim Bruna, Universitat Autònoma de Barcelona, Spain Kenji Fukaya, Kyoto University, Japan Nigel Hitchin, University of Oxford, UK Vaughan Jones, University of California, Berkeley, USA Pierre-Louis Lions, Collège de France, France Gregory Margulis, Yale University, USA Richard Taylor, Harvard University, USA S. R. Srinivasa Varadhan, Courant Institute of Mathematical Sciences, USA Claire Voisin, Institut de Mathématiques de Jussieu, France Enrique Zuazua, Universidad Autónoma de Madrid, Spain
Panels for the program of ICM 2006 1. Logic and Foundations Chair. Angus MacIntyre, University of London, UK. Core Members. Saharon Shelah, Hebrew University, Israel; Hugh Woodin, Univer- sity of California, Berkeley, USA. Other Members. Gregory Cherlin, Rutgers University, USA; Alexander Kechris, California Institute of Technology, USA; Richard Shore, Cornell University, USA; Stevo Todorcevic, University of Toronto, Canada and C.N.R.S., France. The committees of the Congress 17
2. Algebra Chair. Alexander Lubotzky, Hebrew University, Israel. Core Members. Robert Griess, University of Michigan, USA; Vladimir Voevodsky, Institute for Advanced Study, USA. Other Members. William M. Kantor, University of Oregon, USA; Consuelo Martínez López, Universidad de Oviedo, Spain; Dimitry Orlov, Steklov Mathematical Institute, Russia; Idun Reiten, Norwegian University of Science and Technology, Norway. 3. Number Theory Chair. Hendrik Lenstra, Universiteit Leiden, The Netherlands. Core Members. Gerd Faltings, Max-Planck-Institut für Mathematik, Germany; Henryk Iwaniec, Rutgers University, USA; Kazuya Kato, Kyoto University, Japan. Other Members. Haruzo Hida, University of California, Los Angeles, USA; Shou- Wu Zhang, Columbia University, USA. 4. Algebraic and Complex Geometry Chair. Miles Reid, University of Warwick, UK. Core Members. Ngaiming Mok, University of Hong Kong, People’s Republic of China; Shigeru Mukai, Kyoto University, Japan. Other Members. Spencer Bloch, University of Chicago, USA; Fedor Bogomolov, NewYork University, USA; Rahul Pandharipande, Princeton University, USA; Eckart Viehweg, Universität Duisburg-Essen, Germany. 5. Geometry Chair. Gang Tian, Princeton University, USA. Core Members. Frances Kirwan, University of Oxford, UK; François Labourie, Uni- versité de Paris-Sud 11, France; Hiraku Nakajima, Kyoto University, Japan; Leonid Polterovich, Tel Aviv University, Israel. Other Members. Robert Bryant, Duke University, USA; Richard Schoen, Stanford University, USA. 6. Topology Chair. Andrew Casson, Yale University, USA. Core Member. Stephan Stolz, University of Notre Dame, USA. Other Members. Mladen Bestvina, University of Utah, USA; Michael Hopkins, Massachusetts Institute of Technology, USA; Tomotada Ohtsuki, Kyoto University, Japan; Ronald Stern, University of California, Irvine, USA. 7. Lie Groups and Lie Algebras Chair. Joseph Bernstein, Tel Aviv University, Israel. 18 The committees of the Congress
Core Members. Marc Burger, ETH Zürich, Switzerland;Alexander Eskin, University of Chicago, USA; Jean-Loup Waldspurger, Institut de Mathématiques de Jussieu, CNRS, France. Other Members. Pavel Etingof, Massachusetts Institute of Technology, USA; Stephen Kudla, University of Maryland, USA; George Lusztig, Massachusetts In- stitute of Technology, USA. 8. Analysis Chair. Pertti Mattila, University of Helsinki, Finland. Core Members. Boris Kashin, Steklov Institute of Mathematics, Russia; Terence Tao, University of California, Los Angeles, USA. Other Members. Guy David, Université Paris-Sud 11, France; Ronald De Vore, Uni- versity of South Carolina, USA; Hans Martin Reimann, Universität Bern, Switzerland; Yum-Tong Siu, Harvard University, USA. 9. Operator Algebras and Functional Analysis Chair. Gilles Pisier, Texas A&M University, USA. Core Members. Joachim Cuntz, Universität Münster, Germany; Sorin Popa, Uni- versity of California, Los Angeles, USA; Nicole Tomczak-Jaegermann, University of Alberta, Canada. Other Members. Uffe Haagerup, University of Southern Denmark. Denmark. 10. Ordinary Differential Equations and Dynamical Systems Chair. Yakov Sinai, Princeton University, USA. Core Members. John Guckenheimer, Cornell University, USA; Shahar Mozes, Hebrew University, Israel; Jean-Christophe Yoccoz, Collège de France, France; Lai- Sang Young, New York University, USA. Other Members. Giovanni Forni, University of Toronto, Canada; Yulij Ilyashenko, Cornell University, USA; Steklov Mathematical Institute, Russia. 11. Partial Differential Equations Chair. Gilles Lebeau, Université de Nice Sophia Antipolis, France. Core Members. Luis Caffarelli, University of Texas, USA; Sun-Yung Alice Chang, Princeton University, USA; Lawrence Craig Evans, University of California, Berke- ley, USA; Stefan Müller, Max-Planck-Institut für Mathematik, Germany. Other Members. Alberto Bressan, Penn State University, USA; Yoshikazu Giga, Hokkaido University, Japan; Benoît Perthame, École Normale Supérieure, France. 12. Mathematical Physics Chair. Jürg Fröhlich, ETH Zürich, Switzerland. Core Members. Igor Krichever, Columbia University, USA; Gregory Moore, Rutgers University, USA. The committees of the Congress 19
Other Members. Eugene Bogomolny, Université Paris-Sud 11, France; Giovanni Felder, ETH Zürich, Switzerland; Krzysztof Gawedzki, Institut de Physique Théo- rique, ENS-Lyon, France; Sergiu Klainerman, Princeton University, USA; Israel Michael Sigal, University of Toronto, Canada. 13. Probability and Statistics Chair. David Nualart, University of Kansas, Lawrence, USA. Core Members. Terry Lyons, University of Oxford, UK; Terence Speed, University of California, Berkeley, USA. Other Members. Peter Hall,Australian National University, Canberra,Australia; Shi- geo Kusuoka, University of Tokyo, Japan; Michel Ledoux, Université Paul-Sabatier, Toulouse III, France; David Siegmund, Stanford University, USA; Ofer Zeitouni, University of Minnesota, USA. 14. Combinatorics Chair. Gil Kalai, Hebrew University, Israel. Core Members. Jirí Matousek, Charles University, Czech Republic; Richard Stanley, Massachusetts Institute of Technology, USA; Günter Ziegler, Technische Universität Berlin, Germany. Other Members. Peter Cameron, Queen Mary University of London, UK; Andrew Odlyzko, University of Minnesota, USA;Alexander Schrijver, CWI, The Netherlands; Joel Spencer, New York University, USA. 15. Mathematical Aspects of Computer Science Chair. Shafí Goldwasser, Massachusetts Institute of Technology, USA. Core Members. Johan Hastad, KTH, Sweden; Richard Karp, University of Califor- nia, Berkeley, USA; Emo Welzl, ETH Zürich, Switzerland. Other Members. Michael Kearns, University of Pennsylvania, USA; Peter Shor, Massachusetts Institute of Technology, USA; Éva Tardos, Cornell University, USA. 16. Numerical Analysis and Scientific Computing Chair. Alfio Quarteroni, EPFL, Switzerland. Core Members. Wolfgang Dahmen, RWTH Aachen University, Germany; Leslie Greengard, NewYork University, USA; Eitan Tadmor, University of Maryland, USA. Other Members. Albert Cohen, Université Pierre et Marie Curie, France; Lisa Fauci, Tulane University, USA; Tang Tao, Hong Kong Baptist University, People’s Republic of China. 17. Control Theory and Optimization Chair. Jean-Pierre Puel, Université de Versailles, France. Core Members. William Cook, Georgia Institute of Technology, USA; Jorge No- cedal, Northwestern University, USA; Eduardo Sontag, Rutgers University, USA. 20 The committees of the Congress
Other Members. Ruth F. Curtain, University of Groningen, The Netherlands; Petar V. Kokotovic, University of California, Santa Barbara, USA; Steven I. Marcus, Uni- versity of Maryland, USA.
18. Applications of Mathematics in the Sciences Chair. Olivier Pironneau, Université Pierre et Marie Curie, France. Core Members. Ronald Coifman, Yale University, USA; Karl Sigmund, University of Vienna, Austria. Other Members. Jennifer Chayes, University of Washington, USA; David McLaugh- lin, New York University, USA; George C. Papanicolau, Stanford University, USA; Rolf Rannacher, Institut für Angewandte Mathematik, Germany; Endre Süli, Univer- sity of Oxford, UK; Masahisa Tabata, Kyushu University, Japan.
19. Mathematics Education and Popularization of Mathematics Chair. Wilfried Schmid, Harvard University, USA. Core Member. Jill Adler, University of Witwatersrand, South Africa. Other Members. Dan Amir, Tel Aviv University, Israel; Deborah Ball, University of Michigan, USA; Garth Gaudry, University of Melbourne, Australia; Frederick Leung, University of Hong Kong, People’s Republic of China.
20. History of Mathematics Chair. Karen Parshall, University of Virginia, USA. Other Members. Craig Fraser, University of Toronto, Canada; Jeremy G. Gray, Open University, UK; Jan P. Hogendijk, University of Utrecht, The Netherlands; Michio Yano, Kyoto Sangyo University, Japan.
Fields Medal Committee for 2006 Chair John Ball, University of Oxford, UK Members Enrico Arbarello, Università di Roma La Sapienza, Italia Jeff Cheeger, Courant Institute of Mathematical Sciences, USA Donald Dawson, Carleton University, Canada Gerhard Huisken, Max Planck Institute for Gravitational Physics, Germany Curtis T. McMullen, Harvard University, USA Aleksei N. Parshin, Steklov Mathematical Institute, Russia Tom Spencer, Institute for Advanced Study, USA Michèle Vergne, École Polytechnique, France The committees of the Congress 21
Rolf Nevanlinna Prize Committee for 2006 Chair Margaret Wright, Courant Institute of Mathematical Sciences, USA Members Samson Abramsky, University of Oxford, UK Franco Brezzi, Istituto di Matematica Applicata e Tecnologie Informatiche, Italy Gert-Martin Greuel, University of Kaiserslautern, Germany Johan Håstad, KTH Stockholm, Sweden
Carl Friedrich Gauss Prize Committee for 2006 Chair Martin Grötschel, Konrad-Zuse-Zentrum für Informationstechnik, Germany Members Robert E. Bixby, Rice University, USA Frank den Hollander, Eindhoven University of Technology, The Netherlands Stéphane Mallat, École Polytechnique, France Ian Sloan, The University of New South Wales, Australia
Emmy Noether Lecture Committee for 2006 Chair Ragni Piene, Oslo University, Norway Members Christopher Deninger, Westfälische Wilhelms-Universität, Germany Hesheng Hu, Fudan University, China Cathleen Morawetz, Courant Institute of Mathematical Sciences, USA María Eulalia Vares, CBPF, Brazil
Travel Grants Committee Chair John Ball, University of Oxford, United Kingdom Members Hajer Bahouri, Université de Tunis, Tunisia Zhiming Ma, Chinese Academy of Science, Beijing, China Madabusi S. Raghunathan, Tata Institute, India Michael Tsfasman, Russian Academy of Sciences, Moscow, Russia Marcelo Viana, IMPA, Brazil 22 The committees of the Congress
Authorities and members of the IMU Executive Committee and the organizing committees
Other collaborators of the ICM2006 e-program Maria Julià, Rafael Serra Fuster (Agilgroup). ICM2006 Madrid Official Suppliers Audiovisual equipment: AV Medios Building of exhibition and poster boards: DIP Proyectos Catering: MONICO Gourmet Catering at the ICM2006 party: Mariano e Isabel WiFi: Enterasys Press office Clemente Alvárez, Sherezade Álvarez, Álvaro Antón Sancho. Ignacio F. Bayo, Pablo Francescutti, Mario García, Pilar Gil, Lula Gómez, Abelardo Hernández, Concha Muro, Jeff Palmer, Roberto Rubio, Mónica Salomone (Director), Laura Sánchez. Secretariat Teresa López Rodríguez, Itziar Prats Fernández, Magaly Roldán Plumey. Technical secretariat Rocío Aranda, Mireya Arnoso, José Casero, Ana Belén Córdoba, Belén Gómez Aróstegui, Irene Gutiérrez, Paloma Herrero (Director), Antonio Ortiz, Silvia Recio, Carine Sainte-Rose, Celia Teves, Jordi Traveset. List of sponsors 23 List of sponsors
The ICM 2006 is held under the auspices of the International Mathematical Union and the sponsorship of the following public and academic bodies and private companies and foundations.
Public bodies
Ministerio de Educación y Ciencia Ministerio de Asuntos Exteriores y de Cooperación Agencia Española de Cooperación Internacional Ministerio de Cultura Biblioteca Nacional Dirección General de Comunicación y Cooperación Cultural Sociedad Estatal de Conmemoraciones Culturales Comunidad de Madrid Consejería de Educación Instituto Madrileño de Desarrollo Ayuntamiento de Madrid Madrid-Convention Bureau Concejalía de las Artes Centro Cultural Conde Duque Consejo Superior de Investigaciones Científicas Correos Fundación Carolina Fundación Española para la Ciencia y la Tecnología
Academic bodies Universidad Autónoma de Madrid Universitat de Barcelona Universitat Autònoma de Barcelona Universidad de Castilla-La Mancha Universidad Complutense de Madrid Universidad Nacional de Educación a Distancia Universidad de Sevilla
Association for Women in Mathematics Canadian Mathematical Society Real Sociedad Matemática Española Royal Dutch Mathematical Society Sociedad de Estadística e Investigación Operativa Sociedad Española de Matemática Aplicada 24 List of sponsors
Societat Catalana de Matemàtiques Société Mathématique de France Irish Mathematical Society Mathematical Society of Japan National Council of Teachers of Mathematics, USA Sociedad Española de Investigación en Educación Matemática Federación Española de Sociedades de Profesores de Matemáticas Sociedad Española de Historia de las Ciencias y de las Técnicas
Private companies and foundations Enterasys Fundación Pedro Barrié de la Maza Fundación Ramón Areces Fundación Vodafone Grupo SM, Editorial SM ONCE Springer The King Juan Carlos I of Spain Center of N. Y. University in Madrid Opening ceremony
Sir John Ball, President of the International Mathematical Union
Your Majesty, Señor Ruiz Gallardón, Señora Cabrera, Señora Aguirre, Professor Manuel de León, Distinguished guests, Ladies and gentlemen, ¡Bienvenidos al ICM dos mil seis! Welcome to ICM 2006, the 25th International Congress of Mathematicians, and the first ICM to be held in Spain. We offer our heartfelt thanks to the Spanish nation, so rich in history and culture, for its invitation to Madrid. We greatly appreciate that His Majesty King Juan Carlos is honouring mathematics by His presence here today. While celebrating this feast of mathematics, with the many talking-points that it will provide, it is worth reflecting on the ways in which our community functions. Mathematics is a profession of high standards and integrity. We freely discuss our work with others without fear of it being stolen, and research is communicated openly prior to formal publication. Editorial procedures are fair and proper, and work gains its reputation through merit and not by how it is promoted. These are the norms operated by the vast majority of mathematicians. The exceptions are rare, and they are noticed. Mathematics has a strong record of service, freely given. We see this in the time and care spent in the refereeing of papers and other forms of peer review. We see it in the running of mathematical societies and journals, in the provision of free mathematical software and teaching resources, and in the various projects world-wide to improve electronic access to the mathematical literature, old and new. We see it in the nurturing of students beyond the call of duty. This service is exemplified by the tremendous efforts made over the last four years by Spanish mathematicians to bring this Congress to fruition. I propose that we for- mally record our appreciation of their splendid work through electing by acclamation the President of the Local Organizing Committee, Manuel de León, as President of this International Congress. The Scientific Program of the Congress was in the capable hands of an international Program Committee consisting of Noga Alon (Israel, Chair), Douglas Arnold (USA), Joaquim Bruna (Spain), Kenji Fukaya (Japan), Nigel Hitchin (UK), Vaughan Jones (USA), Pierre-Louis Lions (France), Gregory Margulis (USA), Richard Taylor (USA), 26 Opening ceremony
S. R. Srinivasa Varadhan (USA), Claire Voisin(France), Enrique Zuazua (Spain). The International Mathematical Union is most grateful to the members of this committee, and to the many other mathematicians who served on the sectional panels, for their work in putting together a fine program of lectures. Mathematicians do not own mathematics. But among the many millions who use mathematics daily they are distinguished by their constant search for deeper under- standing based on an appreciation of beauty, simplicity, structure and the power of generalization. Yet the lesson of past centuries is that these vital elements in the develop- ment of mathematics require con- stant invigoration by new ques- tions that come from the world about us. There is no object, large or small, and almost no aspect of human existence, to which math- ematics cannot contribute under- standing. In particular, the great questions facing the planet, such as how to model and manage the cli- mate, pose profound mathematical challenges. The need for an un- derstanding of mathematics, of the mathematical way of thinking, and of the role mathematics can play in society, is no longer confined to scientists and engineers, but is in- creasingly important for those who work in industry, finance, the so- cial sciences, and in many other walks of life, and thus also for all involved in ed- ucation, for the media, opinion-formers and politicians. As subjects become better understood, they become more mathematical. Thus in the life sciences, for example, we see a rapid increase in the use of mathematical models, a trend that promises to profoundly influence medicine in the future. In contemplating the importance of mathematics for the world, we see the impor- tance of supporting the development of mathematics throughout the world. Mathe- matical talent does not respect geographical boundaries, but the opportunities, work- ing conditions and tradition necessary for such talent to flourish depend heavily on geography, economic conditions and politics. Each country and region has its own needs for science and mathematics, its own problems as regards its mathematical development. Opening ceremony 27
It is for these reasons that the IMU has made a special effort over the last four years to increase its support for mathematicians in developing countries. It has established an office for developing countries at the International Centre for Theoretical Physics in Trieste, and has cooperated with ICTP and the Abel Fund in the founding of the Ramanujan Prize for young mathematicians working in developing countries. At the IMU General Assembly held in Santiago de Compostela last weekend, a new class of Associate Membership was created to encourage more countries to join the Union. The IMU has augmented its developing countries programmes, particularly in Africa, helped by generous support from the following sponsors: Niels Henrik Abel Memorial Fund (annual grant), Nuffield Foundation and the Leverhulme Trust (linked grants to support math- ematics in sub-Saharan Africa, in conjunction with the London Mathematical Society and the African Mathematics Millennium Science Initiative), David and Lucile Packard Foundation, Andrew W. Mellon Foundation, American Mathematical Society, London Mathematical Society. Other sponsors, including those of the ICM itself, have made it possible for some 400 mathematicians from developing and economically disadvantaged countries, par- ticularly younger researchers, to attend this Congress: ICM sponsors, American Mathematical Society, Mathematical Society of Japan, USA Committee for Mathematics, London Mathematical Society, Het Wiskundig Genootschap Netherlands, Italian Mathematical Union (UMI), German Mathematical Society (DMV), European Mathematical Society. Despite these initiatives, a dramatic increase in both funding and scientific inter- change is required to address the global imbalances in mathematical education and research. In sharing mathematical knowledge and experience with those who work around the world, it is the whole mathematical community that benefits, and we make our own contribution to peace and stability through the binding together of peoples by a common language independent of politics, religion and culture. I wish you all a rewarding and exciting Congress. 28 Opening ceremony
Manuel de León, President of the ICM2006 Organizing Committee Your Majesty, President of the Community of Madrid, Minister of Education and Science, Mayor of Madrid, Professors John Ball and Phillip Griffiths, Dear Colleagues, ladies and gentlemen, On behalf of the Organizing Committee I would like to welcome you all to the ICM2006, and in particular to this opening ceremony. First of all, I want to express our gratitude to the King Juan Carlos for His contin- uous support. ¡Muchas gracias, Majestad! The ICM is a congress of great importance. Every four years, mathematicians from all over the world meet to celebrate mathematics, to inform each other of our latest results, to honour the most outstanding achieve- ments during this period, to debate the present and fu- ture state of the discipline, to discuss how best to trans- fer new knowledge, and to bring mathematics closer to society and to improve pub- lic appreciation. We Spanish mathemati- cians feel very honoured to have been entrusted by the IMU with the organization of this ICM. The constant support of the IMU Execu- tive Committee and its president, Professor John Ball, has been essential for this task. Furthermore, an event of this magnitude requires a great financial commitment, which on this occasion in Madrid has amounted to approximately two and a half million euros. Much of this funding has been provided by the Ministry of Education and Science, the Ministry of Foreign Affairs, the Ministry of Culture, the Community and the City of Madrid, and many others that you can see listed below; we are grateful for that support. In addition to the scientific activities, an interesting series of round-table discus- sions has been organized, as well as an extensive programme of cultural and parallel activities. The fact that the entire process of registration and communication has been carried out electronically is also worthy of mention. Both this opening ceremony Opening ceremony 29 and all the plenary lectures will be transmitted online throughout the world, and the congress records will be available on the website. This is a reflection of the importance of mathematics in today’s Information Society. The logo deserves a special mention because it is an essential part of every ICM, and on this occasion in Madrid it is the basis on which we have built the image of the Congress. It depicts a sunflower consisting of optimum growth mathematics and the golden mean representing a Spain of sunlight and optimism. Our wish and our aim is for this ICM2006 to provide a platform for making the presence of mathematics felt in society, a process in which the media must inevitably play its part. To this end, the ICM2006 set up a Press Office that has worked together with the IMU to issue a weekly bulletin in English and Spanish aimed at both the media and the public at large. Allow me also to explain how Spain has prepared itself for this day. Ten years ago, Spanish mathematicians set about reorganizing the social structure of our community, and in particular the Committee that liaises with the International Mathematical Union. At the same time, we began working on the organization of the World Mathematical Year in Spain. This collective project has had one important consequence: Spanish mathematicians came to the realization that we are a community, a community which, perhaps more than any other scientific discipline in this country, has subjected its strengths and weaknesses to an ongoing process of examination. We have learned that mathematical research in Spain has made great progress in recent years. We now need to raise the standard of quality and encourage the interdisciplinary nature of mathematics, and are presently taking steps to achieve these aims. We all agree on the vital role of mathematics in education, but it still remains to convince everyone of its equally important role as key technology for development. The new schemes for mathematical research currently being set up in Spain will un- doubtedly help to increase qualitative and quantitatively the presence of Mathematics in science, technology and innovation. For a country like ours, this is the basis for a prosperous future. The celebration of the ICM2006 in Madrid constitutes a landmark on this road. It also underlines our sense of belonging to an international community to whose organization and activities we are eager to make a contribution, continuing the process of internationalization that is already under way in Spain. On behalf of the Organizing Committee, I would like to thank all the participants for coming to Madrid, some of them from very distant places, and I apologize if at any time we have been unable to meet all their demands. We would like you to know that in these coming days we are about to share, we are at your complete disposal. Welcome to the ICM2006, welcome to Madrid! We wish you a successful con- ference and a very pleasant stay. Thank you all very much! 30 Opening ceremony
Alberto Ruiz Gallardón, Mayor of Madrid Your Majesty, Under the auspices of the Crown, and in keeping with the scientific and cultural progress that this Institution has enabled in Spain, Madrid bears today the honour and the responsibility of being the world capital of mathematical science. This has been made possible by the International Mathematical Union’s choice of our city to host its twenty-fifth Congress, responding not only to the efforts made by Madrid City Council during its candidacy for the most prestigious mathematical event of our age, but also to the very nature of the capital of Spain as a crossroads of knowledge. Over the next few days, the Congress promoters will witness this city’s unques- tionable capability regarding the organisation of significant events, with international impact, in the fields of science, culture, sports or economics. Madrid’s excellent infrastructures for the stag- ing of fairs and exhibi- tions and the wide range of services it offers are factors which explain the city’s increasingly consoli- dated position at the fore- front in terms of the host- ing of conferences and con- gresses. However, the hos- pitality of the inhabitants of Madrid – on whose behalf I take this opportunity to ex- tend a warm welcome to the participants of the Congress – and their interest in the intellectual progress of this century, are elements that have proved even more decisive in choosing Madrid for a gathering of the brightest minds of our age. Madrid’s long-standing relationship with mathematics can be traced back to at least 1582, when Phillip II founded the Royal Academy of Mathematics. Madrid also provided the stage upon which Agustín de Pedrayes, one of Spain’s most cele- brated mathematicians, who achieved fame in the International Congress of 1799 in Paris via his crucial contribution to the creation of the decimal metric system, car- ried out his professional activity between the 18th and 19th centuries. Nevertheless, Madrid’s confidence in the success of this Congress is focused more on the future than on the past. In a world where political systems appear faced with terrible challenges, where technology is hampered by uncompromising materialism, where the humanities seem overwhelmed by the challenge of providing an answer, you, as mathematicians, have the great privilege of speaking a different and eternally youthful language, wherein conjectures can be tested or refuted, whilst the language itself remains untainted by despair and its universality undiminished by disagreement. Laymen have difficulty Opening ceremony 31 understanding the terms of the debate centred around Fermat’s Last Theorem or those used to describe the essential humility of the mathematical possibilities surrounding Gödel’s Theorem. Nevertheless, we believe that certain equations are as beautiful as the Iliad, as stated by the philosopher Edgar Quinet, and therefore we can find motives for consolation and hope. Indeed, beyond the specific applications of mathematical science, this discipline provides proof that the human need for understanding and for creativity remain intact. In these turbulent and yet complacent times, the inherent difficulty of mathematics – where, in the words of Plato, beauty and truth coincide – constitutes an intellectual stimulus, an invitation to better ourselves and a bastion of purity. The thinker George Steiner quoted Kepler to explain this phenomenon: “amidst massacres [and war], the laws of elliptical motion belong to no man and to no principality”. Therefore, Sire, perhaps it is not only mathematicians who need to come closer to society, by descending to the details of everyday life and explaining the usefulness of their science to the people, but it is all of us who are obliged to make the ethical and aesthetical effort to reach the same level of excellence, rigour and beauty that they inhabit. In this two-fold hope and trust, it is a great pleasure for me to welcome the world’s best mathematicians to Madrid.
Mercedes Cabrera, Minister for Education and Science
Your Majesty, When the International Mathematical Union chose the City of Madrid to host the 25th International Congress of Mathematicians and for the presentation of the pres- tigious Fields Medals, the Spanish Government Ministry of Education and Science understood that it had to give its full support to the Organizing Committee of the event. In this way, our support for the Congress be- came an important part of the Scientific Policy Pro- gramme of Complementary Measures. Given that more than 8,000 Spanish mathemati- cians are represented at the International Mathematical Union, we feel that at this time and together with them all the citizens of Spain are likewise represented, since we recognize the importance of mathematics in the development of thought, in the shaping and management of reality, and in the progress of Culture. 32 Opening ceremony
This Congress will serve to pave the way to new avenues of research, and to the exchange of new advances and opinions among all the different fields of knowledge, whether related directly or indirectly with mathematics. Furthermore, it will assist in advising public bodies, the Ministry of Education and Science among them, on the means of support it should adopt in the future. We therefore celebrate this meeting as a great event of concern to everyone, and with our best wishes for its success we greet the members of the International Math- ematical Union; the Organizing Committee of the Congress; the scientists who will be honoured for their achievements, and all the participants, especially all our guests from abroad. We do not know what some of the pioneers in the history of mathematical re- search in Spain would have said if they had known about the congress we inaugurate today. However, we know what we, as scientists, teachers or holders of public office, are obliged to offer both for them and for society in the matter of teaching and research. For them, in recognition and in tribute to their work and their memory; for society at large, because it is the proper task of public bodies, and in particular of the Ministry I represent, to make scientific achievement and discovery available to all in the interests of progress. That is why the Spanish Ministry of Education and Science is currently developing different projects in support of research in the fields of Mathematical Science. As a basic policy, it is our aim to improve the role of mathematics in Education. To that end, the restructuring of the Spanish educational system as laid out in the new Organic Law of Education will enable us establish specific areas in which the acquisition of logic-mathematical skills and the development of mathematical learning at school level will accorded the status of basic abilities. It is also our aim to pursue a new teaching methodology, encouraging practi- cal experience in the classroom and furthering ongoing teacher training to meet our educational goals. With regard to the situation of research in our country, Spain occupies the 9th po- sition in the world in terms of scientific production in mathematics. This constitutes a considerable improvement over the last twenty-five years, and we expect the plans that are now being developed to improve this performance even further, by consol- idating quality research teams and strengthening relations with researchers working on projects in other fields connected with technological innovation and development. To achieve these goals, the National Programme for Mathematics has been set up within the framework of the R+D+I National Plan. In addition, the Consolider Mathematica Programme is being carried out as part of the broader Consolider Ingenio 2010, which will enable us to fund research work by top level groups in the field of Mathematics. In fact, the Consolider Mathematica programme has been chosen by the relevant Scientific Committee as a worthy recipient of immediate support. We trust that these and other complementary measures, aimed at the training and mobility of research personnel and the creation of research institutions, will strengthen Opening ceremony 33
Mathematical Science, both at an international level and in the Spanish System of Science, Technology and Business. But today we are also here to celebrate the presentation of the Fields Medals and the Nevanlinna and Gauss Prizes. The Ministry of Education and Science of the Spanish Government congratulates the award-winners and expresses its gratitude for their generous contribution to society through Science. They are an example to us all, and especially to the many young people who are laying the foundations for their future professional careers. Finally, I would like to express our gratitude to the members of Executive Com- mittee of the Congress for all their efforts, and to all those involved in the organization of the many accompanying events and activities, as well as the scientists who by their work will enrich discussion and extend common understanding. Your Majesty, we are aware of Your interest in seeing Spain occupy an important place in the society of knowledge and in the International Scientific Community, and of Your wish that all citizens acquire a solid education and training in preparation for the modern world. The Ministry of Education and Science is currently working on numerous schemes and projects to achieve this aim. If we are successful, we will have fulfilled one of the main purposes of our existence. Support for this International Congress of Mathematicians constitutes a further step along this road. Many thanks and welcome to you all.
Esperanza Aguirre, President of the Community of Madrid Your Majesty, It is an honour and a source of great satisfaction for Madrid to welcome from today over 4,000 of the best mathematicians from all over the world who have gathered here to share their latest studies and discov- eries, to explain the state of their researches, and to reward the most outstand- ing achievements of their colleagues with the Fields Medals and the Rolf Nevan- linna and Carl Friedrich Gauss Prizes. In the 19th century, the German mathematician Gustav Jacobi stated that those who devoted themselves to study and research in Mathematics did so above all to “honour the human spirit”". This is how it has always been since the origins of mathematical thought, and the greatest achievements in this Science figure among the finest creations of humanity as a whole. 34 Opening ceremony
This is a good opportunity to recall that mathematical thought dates back to clas- sical Greece, and lies at the source of all Western thought and civilization. From these brilliant beginnings, the essence of mathematical thought has consisted in finding an exact formulation for what we see, what we experience and what we perceive. That is why, whether we realize it or not, we are all descendants of Pythagoras. To these words of greeting and welcome to Madrid I would also like to add my sincere congratulations to all the Congress participants for their efforts, for their researches, and for the knowledge they impart to their students, initiating them into the mysteries of their science. The intrinsic difficulties of their studies sometimes deprive them of recognition by society at large. That is why, with my congratulations, I would like to encourage them to continue with their fine work. In the mid-19th century, the leading Spanish mathematician of the period, José Echegaray, stated that unfortunately there were few Spaniards of world status in mathematical research at that time. Happily today that is no longer the case, and the presence of the most outstanding mathematicians here in Madrid is the best demon- stration of what I mean. I am sure that the celebration of this Congress in Madrid will act as a stimulus to all those young people in Spain who have discovered the pleasures of studying mathematics, and have begun to appreciate the beauty of its reasoning and its proofs. I likewise trust that the choice of Madrid as host city for this Congress will also be a source of pride and encouragement for all the Faculties of Mathematics at Madrid Universities. With my very best wishes for the success of the Congress, and for the personal future and professional scientific career of all those concerned, I once again most cordially welcome you all to Madrid. Many thanks.
Presentation of the new IMU logo by Phillip Griffiths, Secretary of IMU The International Mathematical Union (IMU) has adopted a new logo. It was the win- ner of an international open competition announced by the IMU in 2004 that attracted over 80 submissions. The final selection was made by the Executive Committee of IMU. The logo was designed by John Sullivan, Professor of Mathematical Visualization at the Technical University of Berlin (TU Berlin) and at the DFG Research Center MATHEON, and adjunct professor at the University of Illinois, Urbana (UIUC), with help from Prof. Nancy Wrinkle of Northeastern Illinois University. The logo design is based on the Borromean rings, a famous topological link of three components. The rings have the surprising property that if any one component is removed, the other two can fall apart (while all three together remain linked). This so-called Brunnian property has led the rings to be used over many centuries in many cultures as a symbol of interconnectedness, or of strength in unity. Although the Borromean rings are most often drawn as if made from three round circles, such a construction is mathematically impossible. Opening ceremony 35
The IMU logo instead uses the tight shape of the Borromean rings, as would be obtained by tying them in rope pulled as tight as possible. Mathematically, this is the length-minimizing configuration of the link subject to the constraint that unit-diameter tubes around the three components stay disjoint. This problem and its solution are described in the paper Criticality for the Gehring Link Problem by J. Cantarella, J. Fu, R. Kusner, J. Sullivan, N.Wrinkle, Geometry andTopology10 (2006), pp. 2055–2115, also available at arXiv.org/math/0402212. Although this critical configuration is quite close to one made of convex and concave circular arcs, its actual geometry is surprisingly intricate. Each component is planar and piecewise smooth, with the shapes of many of the 14 pieces described by elliptic integrals. The improvement over the similar piecewise circular configuration leads to a savings of length of less than one tenth of one percent! (The paper cited above first noticed a similar surprise in the simple clasp: one rope attached to the floor clasped around another attached to the ceiling. There as well, the minimizing shapes for the ropes are quite complicated, leaving a small gap between the thick tubes right at the tip.) The tight configuration of the Borromean rings has pyritohedral symmetry (3*2 in the Conway/Thurston orbifold notation), and the IMU logo uses a view along a three-fold axis of rotation symmetry. Instead of the thick tubes, which would touch one another all along their lengths, thinner tubes are drawn, allowing a better view of the link. Sullivan says the new logo “represents the interconnectedness not only of the various fields of mathematics, but also of the mathematical community around the world.” Together with Charles Gunn of TU Berlin, he has made a 5-minute computer- graphics video The Borromean Rings: a new logo for the IMU that was shown at the ICM opening ceremony. 36 Opening ceremony
Presentation of the Fields Medals by John Ball, Chairman of the Fields Medal Committee
The 2006 Fields Medal Committee consisted of: – Enrico Arbarello (Italy) – John Ball (UK, Chair) – Jeff Cheeger (USA) – Donald Dawson (Canada) – Gerhard Huisken (Germany) – Curtis McMullen (USA) – Alexey Parshin (Russia) – Tom Spencer (USA) – Michèle Vergne (France) The instructions to the Committee are: to choose at least two, with a strong preference for four, Fields Medallists, to have regard in its choice to representing a diversity of mathematical fields, and to respect the age limit that a candidate’s 40th birthday must not occur before January 1st of the year of the Congress at which the Fields Medals are awarded. The Committee was privileged to consider a number of remarkable young mathe- maticians. Although the choice was a difficult one, the Committee was unanimous in selecting four medallists whose wonderful work demonstrates the breadth and rich- ness of the subject. I will announce the names of the winners in alphabetical order. A Fields Medal is awarded to Andrei Okounkov, of the Department of Mathemat- ics, Princeton University, for his contributions bridging probability, representation theory and algebraic geometry. A Fields Medal is awarded to Grigory Perelman, of St Petersburg, for his contri- butions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow. I regret that Dr. Perelman has declined to accept the medal. A Fields Medal is awarded to Terence Tao, of the Department of Mathematics, University of California at Los Angeles (UCLA), for his contributions to partial dif- ferential equations, combinatorics, harmonic analysis and additive number theory. A Fields Medal is awarded to Wendelin Werner, of the Laboratoire de Mathéma- tiques, Université Paris-Sud, for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and confor- mal field theory. Opening ceremony 37
Presentation of the Nevanlinna Prize by Margaret H. Wright, Chair of the 2006 Nevanlinna Prize Committee
It is a privilege to announce the winner of the 2006 Rolf Nevanlinna Prize, which is awarded by the International Mathematical Union for outstanding contributions in mathematical aspects of information sciences. The Nevanlinna Prize was first awarded in 1982. A requirement is that the winner’s 40th birthday must occur on or after January 1 of the year in which the award is made. The members of the 2006 Nevanlinna Prize Committee are: – Samson Abramsky (United Kingdom) – Franco Brezzi (Italy) – Gert-Martin Greuel (Germany) – Johan Håstad (Sweden) – Margaret Wright, Chair (United States). The International Mathematical Union awards the 2006 Nevanlinna Prize to Pro- fessor Jon M. Kleinberg of the Computer Science Department, Cornell University, Ithaca, New York, USA. Professor Kleinberg’s date of birth is October 1971. The Nevanlinna Prize citation for Jon Kleinberg is: For deep, creative and insightful contributions to the mathematical theory of the global information environment, including the influential “hubs and authorities” algorithm; methods for discovering short chains in large social networks; techniques for modeling, identifying and analyzing bursts in data streams; theoretical models of community growth in social networks; and contributions to the mathematical theory of clustering. Jon Kleinberg’s combination of mathematical ability, superb taste in interesting problems, breadth of interests and sense of strategy is both dazzling and unmatched. His work has had a fundamental impact on the effectiveness of today’s most ad- vanced Web search engines, and his mathematical insights have had applications to Internet routing, data mining, discrete optimization, and the sociology of the World Wide Web.
Presentation of the Gauss Prize by Martin Grötschel, Chair of the Gauss Prize Committee
Today we celebrate the first award of the Carl Friedrich Gauss Prize for applications of mathematics. Since this is a new IMU distinction, it is appropriate to say a few words about the scope of the prize in the beginning. 38 Opening ceremony
The Gauss Prize is awarded jointly by the Deutsche Mathematiker-Vereinigung (DMV, the German Mathematical Society) and the International Mathematical Union (IMU), and is administered by the DMV.The prize consists of a medal and a monetary award (currently EUR 10,000). The source of the prize fund is a small surplus from the International Congress of Mathematicians (ICM’98) held in Berlin eight years ago. The statutes stipulate that the Gauss Prize is awarded for outstanding mathemat- ical contributions that have found significant practical applications outside of math- ematics, or for achievements that made the application of mathematical methods to areas outside of mathematics possible in an innovative way, e.g., via new modelling techniques or the design and implementation of algorithms. In a nutshell, the Carl Friedrich Gauss Prize is given for the impact the work of the prize winner has had in practice. Since the practical usefulness of mathematical results is often not immediately visible, and as their applicability and importance for practice may only be realized after a long time lag – in contrast to the Fields Medal and the Rolf Nevanlinna Prize – no age limit restricts the choice of a prize winner. Scientific awards gain reputation through the choice of outstanding winners. The Fields Medal, e.g., is a prime example for this fact. The Gauss Prize jury hopes to mark the beginning of a similar tradition today by presenting an awardee whose research has influenced the world at large and whose contributions are highly respected by the mathematical community. Why has the prize been given Gauss’name? Carl Friedrich Gauss (1777–1855) was one of the greatest mathematicians of all time. Gauss combined scientific theory and practice like no other before him or since. His Disquisitiones Arithmeticae, published in 1801, stand to this day as a true masterpiece of scientific investigation. In the same year, Gauss gained fame in wider circles for his prediction, using very few observations, of when and where the asteroid Ceres would next appear. The method of least squares, developed by Gauss as an aid in his mapping of the state of Hannover, is still an indispensable tool for analyzing data. His sextant is pictured on the last series of the German 10-Mark bills, honoring his considerable contributions to surveying. On the front side of the bill, one also finds a bell curve, which is the graphical rep- resentation of the Gaussian normal distribution in probability. Together with Wilhelm Weber, Gauss invented the first electric telegraph. In recognition of his contributions to the theory of electromagnetism, the international unit of magnetic induction is the gauss. These few examples show that the impact of Gauss’ mathematics can be ex- perienced every day everywhere. With this new prize, IMU and DMV would like the world to recognize the importance of mathematics for our society; and Carl Friedrich Gauss is a prime example for the role mathematicians can play in this respect. The medal, designed by Jan Arnold, that comes along with the award is displayed below. It shows on the front a familiar Gauss portrait dissolved into a linear pattern, the least squares method as well as the discovery of Ceres’ orbit are symbolized on the back. Opening ceremony 39
Following the prize statutes, the IMU Executive Committee appointed a Gauss Prize Committee for the 2006 award. The members were Bob Bixby, Martin Grötschel (chair), Frank den Hollander, Stéphane Mallat, and Ian Sloan. The establishment of the Gauss Prize was announced on April 30, 2002, Gauss’ 225th birthday, and at the same time, nominations were invited. About thirty highly deserving mathematicians from all over the world were suggested for this prize by colleagues from pure and applied mathematics. And now, I would like to announce the winner and read the citation. The International Mathematical Union and the Deutsche Mathematiker-Vereini- gung jointly award the Carl Friedrich Gauss Prize for Applications of Mathematics to Professor Kiyosi Itô for laying the foundations of the theory of stochastic differential equations and stochastic analysis. Itô’s work has emerged as one of the major math- ematical innovations of the 20th century and has found a wide range of applications outside of mathematics. Itô calculus has become a key tool in areas such as engi- neering (e.g., filtering, stability, and control in the presence of noise), physics (e.g., turbulence and conformal field theory), and biology (e.g., population dynamics). It is at present of particular importance in economics and finance with option pricing as a prime example. The document is signed by John Ball, President of IMU, and Günter M. Ziegler, President of DMV. A side remark, at this moment in time, a new application-oriented research insti- tute called Quantitative Products Laboratory, is being founded in Berlin. It will be sponsored by a big German bank donating at least three million Euros annually. This new institute would not have been founded without the foundations laid by Kiyosi Itô. The details of Itô’s work will be explained tomorrow, August 23, in the Gauss Prize lecture presented by Hans Föllmer. Kiyosi Itô was born in Japan in 1915. He received his Doctor of Science degree from the Imperial University in Tokyo and is now professor emeritus at Kyoto Uni- versity. He has honorary doctoral degrees from Université Paris VI, ETH Zürich, and the University of Warwick. Itô is a member of the Académie des Sciences, France, 40 Opening ceremony the Japan Academy, and the National Academy of Sciences, USA, to mention just a few of his many honors and distinctions. For health reasons, Kiyosi Itô is unfortunately unable to be present at today’s award ceremony.
The IMU President, John Ball, will personally take the Gauss Medal to Kyoto after this meeting and present it to Professor Itô in a special ceremony1. I am very happy to be able to announce that the Itô family has decided to send a representative to Madrid. Kiyosi Itô’s youngest daughter, Junko Itô, who is professor and chair of linguistics at the University of California in Santa Cruz, is here to accept the Gauss Prize on behalf of her father.
1Photograph by Armin Mester. It shows John Ball and Kiyosi Itô on September 14, 2007 in Kyoto during the presentation of the Gauss Medal. Opening ceremony 41
His Majesty the King of Spain, Juan Carlos
I am greatly pleased to preside over the opening of this Twenty-Fifth International Congress of Mathematicians, an outstanding scientific event which, in addition to its tradition of over a century, enjoys unquestionable prestige and significance on a global scale. I extend my greetings to all the participants, my warm welcome to Spain to those from other countries, and my most heartfelt congratulations to the organizers of this Congress in Madrid. You will understand that it is a very special pleasure for me that this Congress, which has brought together nearly four thousand scientists from over one hundred countries, is being held for the first time in our country. Therefore, I wish to con- vey my greatest recogni- tion and appreciation to the Spanish mathematical com- munity, whose well-deserved prestige, proven effort and cohesion, have made Spain – and more specifically Madrid – the focus of attention of the international mathemati- cal community this year. This Congress enables us to learn about the main progress made by research in this discipline, and to high- light and promote in our re- spective societies the enor- mous importance of Mathematics. Importance because it is a basic instrument to understand the world, because it constitutes an unquestionable pillar of education, and because it is an indispensable tool to ensure progress for the benefit of Humanity. Galileo told us that the world is written in mathematical language; to understand it, nothing can rival this discipline that has brought us together today in Madrid. Improved understanding of the world we live in, by using the universality of Mathematics, is, in addition, a task which reinforces cooperation between diverse countries, societies and cultures. It is equally evident that the high value of Mathematics in education requires our attention and dedication. Mathematics is rightly considered the key technology. This is stated in the Decla- ration made public in 2000 by the International Mathematical Union and UNESCO, on the occasion of the World Mathematical Year. 42 Opening ceremony
We depend on, and we will increasingly depend on, the indispensable foundation that research, technology and innovation constitute for the future of our economic development and our social well-being. For this reason, we must promote mathematical development as an essential ele- ment for progress that will enable sustainable development for all Humanity. The business world must also join, with increasing efforts, a discipline which has been essential to our development, as it is, for example, the basic foundation to achieve the Information Society we currently enjoy. Spain has been making a particular effort, and will continue to do so in the future, to promote its technological development. Projects such as the Ingenio 2010 Programme are aimed at such a goal. We are pleased to see that Spanish mathematicians have not missed the opportunity to participate in such an ambitious R+D+I programme. But this Congress also includes other aspects that I would like to highlight. With 108 countries represented, the largest number in its history, it aims to achieve universal representation and participation. This has been made possible thanks to grant programmes for the participation of mathematicians from economically disadvantaged countries, following a long- standing tradition of the International Mathematical Union to which Spain is especially sensitive. Furthermore, this Congress has made an enormous effort to bring Mathematics closer to society, striving to make it more wide-spread and well-known in public opinion. All of this is being done through exhibitions, various cultural events and by rein- forcing its presence in the media. This effort to make Mathematics more well-known is particularly significant, as it is fundamental to encourage new scientific vocations all over the world. In addition, these Congresses enable the mathematical community to award, ev- ery four years, and with well-deserved solemnity, its most highly prized and valued distinctions. I am referring to the Fields Medal, the Rolf Nevanlinna Prize and the Carl Friedrich Gauss Prize, all of which are indisputably prestigious awards which have just been granted to this year’s winners. The Fields Medal has been awarded for 70 years to mathematicians under the age of 40 for outstanding achievement in the basic aspects of the discipline; the Rolf Nevanlinna Prize has been awarded since 1982 for the best mathematical contributions to the Information Society; and the Carl Friedrich Gauss Prize, awarded for the first time this year in Madrid, aims to honour outstanding achievement in contributing to improve our everyday lives. I extend my most enthusiastic congratulations to all this year’s winners. Their work, professional career and scientific merits, as well as their contribution to our societies’development and well-being, deserve everyone’s recognition and con- stitute an example and encouragement for the whole of the international mathematical Opening ceremony 43 community. To conclude, I would like to reiterate my most sincere support for the significant work done by the International Mathematical Union. I wish you every success for the next Congress to be held in India, just as I trust this one in Madrid will be successful. I declare open the 25th International Congress of Mathematicians of 2006. Thank you very much.
The King of Spain, Juan Carlos. On his left, Jon Kleinberg and Terence Tao; on his right Andrei Okounkov and Wendelin Werner.
Closing ceremony
The closing ceremony was held on Wednesday, August 30, starting at 18:00 in Audi- torium A of the Palacio Municipal de Congresos de Madrid.
Sir John Ball, President of the International Mathematical Union Welcome to the Closing Ceremony of ICM 2006! You just saw a longer version of the video that was shown at the Opening Ceremony, concerning the new IMU logo, designed by John Sullivan. It has been a marvellous International Congress, and we begin by a retrospective look at some of the highlights. (Showing of video montage of scenes from the ICM.) Before saying a few words about this Congress, I want to go back to before King Juan Carlos opened ICM 2006, and present a brief report about the IMU General Assembly held in Santiago de Compostela on 19 and 20 August. This was a very successful meeting, which owed much to the care and consideration of the local organizing committee in Santiago chaired by Juan Viaño. The General Assembly voted to make various changes to the Statutes and the Procedures for Election of IMU. In particular a new independent Nominating Com- mittee structure for the construction of slates for elections was approved. In future the Executive Committee of the International Commission on Mathematical Instruc- tion, concerned with mathematics education, will be elected by the ICMI General Assembly rather than the IMU General Assembly. The number of members-at-large on the IMU Executive Committee is increased from five to six, and a new category of Associate Membership of IMU, with no dues, no votes and limited duration, was introduced to encourage full membership. Turning to the elections themselves, the new Executive Committee of IMU to serve for the period 2007–2010 will be: President. László Lovász (Hungary) Secretary. Martin Grötschel (Germany) Vice Presidents. Zhiming Ma (China), Claudio Procesi (Italy) Members at Large. Salah Baouendi (USA), Manuel de León (Spain), Ragni Piene (Norway), Cheryl Praeger (Australia), Victor Vassiliev (Russia), Marcelo Viana (Brazil) I would like to pay tribute to the work of this committee, and especially to the retiring members: Phillip Griffiths, who has completed two terms as Secretary of IMU, Jean-Michel Bismut (Vice-President), Masaki Kashiwara (Vice-President), Jacob Palis (Past-President and Past Secretary of IMU), and M. S. Raghunathan. 46 Closing ceremony
And we also will say farewell to Linda Geraci, who has served admirably as the IMU Administrator. The membership of the Commission on Development and Exchanges, concerned with developing countries, for 2007–2010 will be: President. S.G. Dani (India) Secretary. Gérard González-Sprinberg (France) Members at Large. Graciela Boente (Argentina), Paulo Cordaro (Brazil), Jean-Pierre Gossez (Belgium), Mary Teuw Niane (Sénégal), Marta Sanz-Solé (Spain), Jiping Zhang (China) CDE will be unified with the recently formed Developing Countries Strategy Group (DCSG) to form a new IMU Commission for Developing Countries which will both consider strategy and administer the IMU grants programmes. I would like to thank the members of CDE and in particular Herb Clemens, the outgoing CDE Secretary and the Chair of DCSG, for their excellent work, in which they have been ably assisted by the Developing Countries Administrator Sharon Laurenti. The new Executive Committee of ICMI for 2007–2009 will be: President. Michèle Artigue (France) Secretary-General. Bernard Hodgson (Canada) Vice Presidents. Jill Adler (South Africa), Bill Barton (New Zealand) Members at Large. Maria Bartolini Bussi (Italy), Jaime Carvalho e Silva (Portugal), Celia Hoyles (UK), S. Kumaresan (India), Alexei Semenov (Russia) The fact that this Executive Committee will hold office for three rather than four years is related to the transition process towards the new electoral system for ICMI. The General Assembly also elected two members to the International Commission for the History of Mathematics, Christian Houzel (France) and Peter Neumann (UK). Although it is appointed rather than elected, I want to show you the membership of the Committee for Electronic Information and Communication (CEIC) for the next two years: Chair. Jonathan Borwein (Canada) Members at Large. Michael Doob (Canada), David Eisenbud (USA), John Ewing (USA), Ulf Rehmann (Germany), Alf van der Poorten (Australia) and one member from the IMU Executive Committee. This is one of IMU’s most important committees. Examples of its fine work are the new Electronic and Federated World Directories of Mathematicians. If you have not used these important resources you can learn about them on the IMU webpages. IMU is very grateful to this committee and in particular to its Chair Jonathan Borwein. An important discussion in Santiago concerned IMU’s finances. As well as a 5% increase in dues for each of the next 4 years, an increase in the number of units paid by the generally wealthier countries in groups IV and V was agreed. These increases, Closing ceremony 47 though painful, are essential to pay for IMU’s increased activities, especially with respect to developing countries. A further important discussion concerned a new set of guidelines for the scientific programme of future ICMs. This is a quite lengthy and detailed document, that after revision will be made available on the IMU webpages. The new guidelines begin with a description of the purpose of the ICM: Every ICM should reflect the current activity of mathematics in the world, present the best work being carried out in all mathematical subfields and different regions of the world, and thus point to the future of mathematics. The invited speakers at an ICM should be mathematicians of the highest quality who are able to present current research to a broad mathematical audience. I think that this is an important statement which should help future Program Com- mittees in the difficult job of choosing a geographically balanced list of speakers of the highest quality. The General Assembly passed 11 resolutions. I want to show you the four reso- lutions over which there was some discussion. Resolution 8 concerned mathematical education: The General Assembly of the IMU reaffirms the importance of the issues treated by ICMI (the International Commission on Mathematical Instruction). It recognizes the importance of continuing and strengthening the relationship of IMU with ICMI and urges the increased involvement of research mathematicians in mathematical education at all levels. Resolution 9 concerns CEIC: With the ultimate goal of creating an enduring network of digital mathematical literature, the General Assembly of the IMU endorses the new version of the “Best practices” document of its Committee on Electronic Information and Communication (CEIC), posted June 2005 at http://www.ceic.math.ca, as well as the March 2005 draft of “Digital Mathematical Library: a vision for the Future”. The digital mathematical library is a very important project that we need to do as much as we can to further. Resolution 10 concerns the freedom of movement of scientists and mathemati- cians: The General Assembly of the IMU continues to endorse the principle of univer- sality expressed in the International Council for Science (ICSU ) ARTICLE 5 of the STATUTES, as adopted by the 1998 General Assembly, and endorses the additional ICSU Statement on the Universality of Science (2004). Notwithstanding heightened tensions, security concerns, etc., the General Assembly urges free exchange of scien- tific ideas and free circulation of scientists and mathematicians across international borders. The IMU opposes efforts by governments to restrict contacts, interactions, access and travel in the world mathematical community, particularly when such re- strictions penalize individual mathematicians for the actions of governments. 48 Closing ceremony
Resolution 7 concerns the finances of IMU: The General Assembly recommends that the incoming Executive Committee of the IMU studies the establishment of stable administrative structure and funding mechanisms, including possible fund-raising, for the support of the expanding IMU activities, and report to the 2010 General Assembly with concrete proposals. Finally, the General Assembly decided that the location of ICM2010 will be Hyderabad in India. If you want to learn more about IMU, you can consult the IMU webpages, where a detailed report of the General Assembly will appear, and read the new electronic newsletter IMU-Net. Mireille Chaleyat-Maurel has done a splendid job in the pro- duction of the newsletter – can we express our thanks to her. The planning and bringing together of the many elements that make up the In- ternational Congress is a daunting undertaking. In several respects, such as online internet transmission of the plenary lectures and the management of relations with the media, of which I will have more to say in a moment, ICM 2006 has set standards for the future. In addition, the local organizing committee complemented the scien- tific programme with a tapestry of interesting events and exhibitions, expressing the richness of mathematics through discourse, history and art. To all those who have lived the Congress over the last few years, and to those who have helped during the Congress itself, together making it such a great occasion, we say that your hard work has really been worth it, and how very much it is appreciated by all who have spent these days in Madrid. In his closing address, Manuel de León will give us the opportunity to recognize the many individuals who have contributed to the success of this Congress. But with his permission I want to say a few words about the ICM and the media. This was a cooperative effort between IMU and the ICM Press Office. The Congress turned out to be a remarkable news story, and it was remarkably told. I wish to thank Allyn Jackson and Christof Poeppe, who wrote the initial press releases for the prizes, Marcus du Sautoy, who gave invaluable advice, and through articles and interviews contributed greatly to generating media interest, and Anne-Marie Astad for making available distribution lists developed for the Abel Prize. But no praise is too great for the accomplishments of the ICM Press Office itself, which consisted of Monica Salomone and Ignacio Fernandez Bayo of DIVULGA, supported by the splendid team that you see listed in a section of this volume. The result of their untiring and professional work was unprecedented national and international press coverage of the Congress and of mathematics. Let me end by saying that it has been a great privilege to serve as President of IMU, and to work with my colleagues on the Executive Committee and with the Local Organizing Committee of the Congress. The IMU is very fortunate to have László Lovász as its next President. I wish him every success for his term of office, and thank you all for your participation in the Congress. I now invite László Lovász to address the Congress. Closing ceremony 49
László Lovász, Elected President of the International Mathematical Union Ladies and gentlemen, Let me start with joining John Ball in expressing my sincere thanks and most heart- felt congratulations to the Organizers of the Congress. They have done a tremendous job, and we all benefited from this a lot: not only the participants, but also those colleagues and students to whom we go back and to whom we’ll communicate what we have learned here. I would also like to extend these thanks and congratulations to the Program Com- mittee and the Executive Committee, who also worked very hard over the last 4 years. In particular, I express my thanks to John Ball for his devout, selfless, and I must say, very successful job he did as the President of IMU. It will very difficult to measure up to his work; one fact that helps me face this task is that as Past President, he’ll be a member of the Executive Committee, and I’ll count on his advice and help. I’d also extend these thanks to the retiring members of the Executive Committee: Vice Presidents Jean-Michel Bismut and Masaki Kashiwara, Secretary Phillip Griffiths, Madabusi Raghunathan and Past President Jacob Palis. To those members of the Executive Committee who stayed on to serve a second term, Vice president Zhi-Ming Ma, Secretary Martin Grötschel, Ragni Piene and Victor Vassiliev, I am thankful for their willingness to do so, and I am looking forward to working with them. I also want to express my thanks to the speakers. To be invited to the Congress is a great honor but also a great responsibility. Some areas are easier to talk about to a general mathematical audience than others; but I feel that all our speakers made a great afford to convey the main ideas and results to us. Let me add a few more personal thoughts. When one arrives at a Congress, one cannot feel but overwhelmed by the number of people and by the variety of mathematics that is presented here. One could walk the corridors for minutes without seeing a familiar face, and one could browse the abstracts long before seeing a topic that one, say, did research in. This is so even for a senior person who attended many previous Congresses, and obviously a young person who has not been to previous Congresses must feel this even more. It is perhaps because of this feeling that people repeatedly bring up the idea of abandoning these International Congresses. I feel this would be a serious mistake. I talked to scientists working in other fields, and they expressed their envy for the fact that we have a meeting where the best mathematicians tell to all of us what are the main problems, trends, or paradigms of their fields; where we honor the recipients of major prizes, and hear and discuss their work; where we have panel discussions and also corridor discussions about important issues facing our science or our community. I hope that now, 9 days after the opening, all participants, in particular our young colleagues, go home with a feeling that mathematics is a vibrant, live, beautiful and fruitful science, and this will help them in their research, teaching and popularization of mathematics. And I hope that you’ll come back to the next congress. See you at ICM 2010! 50 Closing ceremony
Manuel de León, President of the ICM2006 Organizing Committee Dear colleagues, Ten days ago in this same auditorium, we opened the 25th International Congress of Mathematicians. Since then we have been meeting here in this impressive Palacio Municipal de Congresos. We hope that this time together has been fruitful, and that you have met old friends and made new ones. The best way of knowing if an ICM has been successful is if the participants are sorry to depart. If that is the case, then remember that it is not “Goodbye” but “See you soon!”, because the great mathematical family will continue to meet at other congresses all over the World, and in four years time we will all be together again in Hyderabad, India, to enjoy the hospitality of our Indian friends. No ICM would be possible without the effort of many people, and now is the moment to acknowledge them. First, those in the different committees: – Local Program Committee – Satellite conferences –Web – Grants – Cultural activities – Social activities – Infrastructure and logistics – Publications The efforts of the Secretariat, under the direction of our General Secretary José Luis González-Llavona, has been fundamental. I hope they will forgive us for any moments of impatience or bad moods during the run-up to the Congress. The work of our congress agency, UNICONGRESS, has also been essential. We have worked together through thick and thin, but the final result has made it all worthwhile. Our thanks to them, to all our suppliers, and to all the staff at the Palacio Municipal de Congresos. I believe that if one thing stands out in this ICM2006, it is the extraordinary coverage provided by the press, and for that we have mainly to thank our friends at DIVULGA, led by Ignacio F. Bayo and Mónica Salomone. Their example is one to be followed, to continue the effort of putting mathematics across to the people. Another crucial help was provided by our over 350 volunteers. Their patient and enthusiasm have contributed to do this ICM unforgettable for all of us. Thank you very much to all of you! Finally, we express our thanks to all the participants for taking the trouble to come to Madrid in such difficult times, to set this great example of tolerance and peaceful co-existence. Thanks to all of you. Have a safe journey home, and see you again in India in 2010! See you all soon! ¡Hasta pronto! Closing ceremony 51
Rajat Tandon, University of Hyderabad, India
Sir John Ball, Professor Lovász, Professor Manuel de Léon, Ladies and gentleman,
Let me first take this opportunity to express my appreciation of the innumerable number of people who have worked so tirelessly for this Congress in Madrid. Let me say ‘Gracias’ to our Spanish hosts and the local organizing committee under the chairmanship of Prof. Manuel de Léon whose monumental effort has ensured the unqualified success of this ICM. I say ‘Gracias’ to the hundreds of volunteers who have been so gracious in rendering their assistance to us. And finally I thank the various committees of the IMU who have presented us with such a strong and exciting academic programme. We recognize that we have our work cut out for us if we are to emulate the success of this Congress in Hyderabad. We in India feel very privileged to have the honour of hosting the next International Congress of Mathematicians. It gives us enormous pleasure to invite the mathematical community from all six continents to Hyderabad for the ICM2010, to be held from the 19th to the 27th of August.
Hyderabad, like Madrid, is a wonderful composition of the old and the new. This city, founded more than 400 years ago, houses teeming bazaars, old jewelry and fine craftsmen, old forts and mausoleums. Cosmopolitan in its population you find people of all faiths living and learning together here. 52 Closing ceremony
Two hundred years ago this city expanded to the twin cities of Hyderabad and Secunderabad with the addition of a cantonment area and today greater Hyderabad is a conglomeration of three cities in one with the modern Cyberabad area which is second only to Bangalore as the information technology heart of India. Here you find not only large research and development centres of the top Indian IT companies like the Tata Consultancy Services or Infosys but also the large multinationals like IBM, Microsoft and Google. This is the charm of Hyderabad – whilst you will find computer scientists at an international institute of information technology grapple with the intricacies of P = NP, you will also find the finest pearl craftsmen in the world – their craft inherited from their forefathers over hundreds of years. Hyderabad is known as the pearl capital of India and perhaps of the world. The organizing committee for the ICM2010 will be pushing for several satellite conferences in different parts of India – north, south, east and west. So those who wish to visit the Taj Mahal or the Pink city of Jaipur or the temples of Mahabalipuram will always be able to find a satellite conference of their choice near the place they want to visit. We are urging other South Asian countries to hold satellite conferences as well. It will be our pleasure to host a meeting of the General Assembly of the IMU in Bangalore on the 16th and 17th of August just prior to the ICM. I urge all delegates here to let it be known to the mathematical community of their countries that an open and democratic India, the home of Ramanujan, with a vibrant community of scholars of its own warmly welcomes them all and urges them to mix mathematics with pleasure and flavour the traditional hospitality of India in the August of 2010. We assure you that it will be a memorable experience.
The granite sculpture created by Keizo Ushio – a beautiful infinity, and the special stamp of the Congress. The work of the Fields Medalists, the Rolf Nevanlinna Prize Winner and the Gauss Prize Winner
Giovanni Felder The work of Andrei Okounkov ...... 55 John Lott The work of Grigory Perelman ...... 66 Charles Fefferman The work of Terence Tao ...... 78 Charles M. Newman The work of Wendelin Werner ...... 88 John Hopcroft The work of Jon Kleinberg ...... 97 Hans Föllmer On Kiyosi Itô’s work and its impact ...... 109
Wendelin Werner, Andrei Okounkov, Terence Tao, Jon Kleinberg, and Junko Itô, youngest daughter of Kiyosi Itô. 54
Andrei Okounkov
B.S. in Mathematics, Moscow State University, 1993 Ph.D. in Mathematics, Moscow State University, 1995 Positions until today
1994–1995 Research Fellow in the Dobrushin Mathematical Laboratory at the Institute for Problems of Information Transmission, Russian Academy of Sciences 1996 Member in the Institute of Advanced Study in Princeton 1997 Member in the Mathematical Sciences Research Institute in Berkeley, on leave from University of Chicago 1996–1999 L. E. Dickson instructor in Mathematics, University of Chicago 1998–2001 Assistant Professor of Mathematics, University of California at Berkeley 2001–2002 Professor of Mathematics, University of California at Berkeley Present position Professor of Mathematics, Princeton University The work of Andrei Okounkov
Giovanni Felder
Andrei Okounkov’s initial area of research is group representation theory, with par- ticular emphasis on combinatorial and asymptotic aspects. He used this subject as a starting point to obtain spectacular results in many different areas of mathematics and mathematical physics, from complex and real algebraic geometry to statistical mechanics, dynamical systems, probability theory and topological string theory. The research of Okounkov has its roots in very basic notions such as partitions, which form a recurrent theme in his work. A partition λ of a natural number n is a non-increasing sequence of integers λ1 ≥ λ2 ≥···≥0 adding up to n. Partitions are a basic combi- natorial notion at the heart of the representation theory. Okounkov started his career in this field in Moscow where he worked with G. Olshanski, through whom he came in contact with A. Vershik and his school in St. Petersburg, in particular S. Kerov. The research programme of these mathematicians, to which Okounkov made substantial contributions, has at its core the idea that partitions and other notions of representa- tion theory should be considered as random objects with respect to natural probability measures. This idea was further developed by Okounkov, who showed that, together with insights from geometry and ideas of high energy physics, it can be applied to the most diverse areas of mathematics. This is an account of some of the highlights mostly of his recent research. I am grateful to Enrico Arbarello for explanations and for providing me with very useful notes on Okounkov’s work in algebraic geometry and its context.
1. Gromov–Witten invariants
The context of several results of Okounkov and collaborators is the theory of Gromov– Witten (GW) invariants. This section is a short account of this theory. GW invariants originate from classical questions of enumerative geometry, such as: how many ra- tional curves of degree d in the plane go through 3d − 1 points in general position? A completely new point of view on this kind of problems appeared at the end of the eighties, when string theorists, working on the idea that space-time is the product of four-dimensional Minkowski space with a Ricci-flat compact complex three-fold, came up with a prediction for the number of rational curves of given degree in the x5 +···+x5 = CP 4 quintic 1 5 0in . Roughly speaking, physics gives predictions for differential equations obeyed by generating functions of numbers of curves. Solving these equations in power series gives recursion relations for the numbers. In particular a recursion relation of Kontsevich gave a complete answer to the above question on rational curves in the plane.
Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2007 European Mathematical Society 56 Giovanni Felder
In general, Gromov–Witten theory deals with intersection numbers on moduli spaces of maps from curves to complex manifolds. Let V be a nonsingular projec- tive variety over the complex numbers. Following Kontsevich, the compact moduli space Mg,n(V, β) (a Deligne–Mumford stack) of stable maps of class β ∈ H2(V ) is the space of isomorphism classes of data (C, p1,...,pn,f)where C is a complex projective connected nodal curve of genus g with n marked smooth points p1,...,pn and f : C → V is a stable map such that [f(C)]=β. Stable means that if f maps an irreducible component to a point then this component should have a finite automor- phism group. For each j = 1,...,ntwo natural sets of cohomology classes can be ∗ ∗ defined on these moduli space: 1) pull-backs evj α ∈ H (Mg,n(V, β)) of cohomology ∗ classes α ∈ H (V ) on the target V by the evaluation map evj : (C, p1,...,pn,f)→ 2 f(pj ); 2) the powers of the first Chern class ψj = c1(Lj ) ∈ H (Mg,n(V, β)) of the L (C, p ,...,p ,f) T ∗ C C line bundle j whose fiber at 1 n is the cotangent space pj to at pj . The Gromov–Witten invariants of V are the intersection numbers τ (α )...τ (α )V = ψkj ∗α . k1 1 kn n β,g j evj j Mg,n(V,β)
If all ki are zero and the αi are Poincaré duals of subvarieties, the Gromov–Witten invariants have the interpretation of counting the number of curves intersecting these subvarieties. As indicated by Kontsevich, to define the integral one needs to con- struct a virtual fundamental class, a homology class of degree equal to the “expected dimension”
vir dim Mg,n(V, β) =−β · KV + (g − 1)(3 − dim V)+ n, (1) where KV is the canonical class of V . This class was constructed in works of Behrend– Fantechi and Li–Tian. The theory of Gromov–Witten invariants is already non-trivial and deep in the case where V is a point. In this case Mg,n = Mg,n({pt}) is the Deligne–Mumford moduli space of stable curves of genus g with n marked points. Witten conjectured and Kontsevich proved that the generating function ∞ n 1 pt F(t ,t ,...)= τk ...τk g tk , 0 1 n! 1 n j n=0 k1+···+kn=3g−3+n j=1 involving simultaneously all genera and numbers of marked points, obeys an infinite set of partial differential equations (it is a tau-function of the Korteweg–de Vries integrable hierarchy obeying the “string equation”) which are sufficient to compute all the intersection numbers explicitly. One way to write the equations is as Virasoro conditions F Lk(e ) = 0,k=−1, 0, 1, 2 ..., for certain differential operators Lk of order at most 2 obeying the commutation relations [Lj ,Lk]=(j − k)Lj+k of the Lie algebra of polynomial vector fields. The work of Andrei Okounkov 57
Before Okounkov few results were available for general projective varieties V and they were mostly restricted to genus g = 0 Gromov–Witten invariants (quantum cohomology). For our purpose the conjecture of Eguchi, Hori and Xiong is rele- vant here. Again, Gromov–Witten invariants of V can be encoded into a generating function FV depending on variables tj,a where a labels a basis of the cohomology of V . Eguchi, Hori and Xiong extended Witten’s definition of the differential oper- F ators Lk and conjectured that FV obeys the Virasoro conditions Lk(e V ) = 0 with these operators.
2. Gromov–Witten invariants of curves
In a remarkable series of papers ([10], [11], [12]), Okounkov and Pandharipande give an exhaustive description of the Gromov–Witten invariants of curves. They prove the Eguchi–Hori–Xiong conjecture for general projective curves V , give explicit descriptions in the case of genus 0 and 1, show that the generating function for V = P1 is a tau-function of the Toda hierarchy and consider also in this case the C×-equivariant theory, which is shown to be governed by the 2D-Toda hierarchy. They also show that GW invariants of V = P1 are unexpectedly simple and more basic than the GW invariants of a point, in the sense that the latter can be obtained as a limit, giving thus a more transparent proof of Kontsevich’s theorem. A key ingredient is the Gromov–Witten/Hurwitz correspondence relating GW in- variants of a curve V to Hurwitz numbers, the numbers of branched covering of V with given ramification type at given points. A basic beautiful formula of Okounkov and Pandharipande is the formula for the stationary GW invariants of a curve V of genus g(V ), namely those for the Poincaré dual ω of a point: λ 2−2g(V ) n p (λ) τ (ω)...τ (ω)• V = dim ki +1 . k1 kn β=d·[V ],g (2) d! (ki + 1)! |λ|=d i=1 The (finite!) summation is over all partitions λ of the degree d and dim λ is the dimension of the corresponding irreducible representation of Sd . The genus g of the domain is fixed by the condition that the cohomological degree of the integrand is equal to the dimension of the virtual fundamental class. It is convenient here to include also stable maps with possibly disconnected domains and this is indicated by the bullet. The functions pk(λ) on partitions are described below. Hurwitz numbers can be computed combinatorially and are given in terms of representation theory of the symmetric group by an explicit formula of Burnside. If the covering map at the ith point looks like z → zki +1, i.e., if the monodromy at the ith point is a cycle of length ki + 1, the formula is λ 2−2g(V ) n H V (k + ,...,k + ) = dim f (λ). d 1 1 n 1 d! ki +1 |λ|=d i=1 58 Giovanni Felder
Thus in this case the GW/Hurwitz correspondence is given by the substitution rule fk+1(λ) → pk+1(λ)/(k + 1)!. The functions fk and pk are basic examples of shifted symmetric functions, a theory initiated by Kerov and Olshanski, and the results of Okounkov and Pandharipande offer a geometric realization of this theory. A shifted symmetric polynomial of n variables λ1,...,λn is a polynomial invariant under the action of the symmetric group given by permuting λj − j. A shifted symmetric function is a function of infinitely many variables λ1,λ2 ...,restricting for each n to a shifted symmetric polynomial of n variables if all but the first n variables are set ∗ to zero. Shifted symmetric functions form an algebra = Q[p1,p2,...] freely generated by the regularized shifted power sums, appearing in the GW invariants: p (λ) = λ − j + 1 k − −j + 1 k + ( − −k)ζ(−k) k j 2 2 1 2 j
The second term and the Riemann zeta value “cancel out” in the spirit of Ramanujan’s + + +···=− 1 f (λ) second letter to Hardy: 1 2 3 12 . The shifted symmetric functions k appearing in the Hurwitz numbers are central characters of the symmetric groups Sn: f1 =|λ|= λi and the sum of the elements of the conjugacy class of a cycle of length k ≥ 2 in the symmetric group Sn is a central element acting as fk(λ) times the identity in the irreducible representation corresponding to λ. The functions pk and fk are two natural shifted versions of Newton power sums. In the case of genus g(V ) = 0, 1, Okounkov and Pandharipande reformulate (2) in terms of expectation values and traces in fermionic Fock spaces and get more explicit descriptions and recursion relations. In particular if V = E is an elliptic curve the generating function of GW invariants reduces to the formula of Bloch and Okounkov [1] for the character of the infinite wedge projective representation of the algebra of polynomial differential operators, which is expressed in terms of Jacobi theta functions. As a corollary, one obtains that qd τ (ω)...τ (ω)• E k1 kn d d belongs to the ring Q[E2,E4,E6] of quasimodular forms. A shown by Eskin and Okounkov [2], one can use quasimodularity to compute the asymptotics as d →∞of the number of connected ramified degree d coverings of a torus with given monodromy at the ramification points. By a theorem of Kontsevich– Zorich and Eskin–Mazur, this asymptotics gives the volume of the moduli space of holomorphic differentials on a curve with given orders of zeros, which is in turn related to the dynamics of billiards in rational polygons. Eskin and Okounkov give explicit formulae for these volumes and prove in particular the Kontsevich–Zorich conjecture that they belong to π −2gQ for curves of genus g. The work of Andrei Okounkov 59
3. Donaldson–Thomas invariants
As is clear from the dimension formula (1) the case of three-dimensional varieties V plays a very special role. In this case, which in the Calabi–Yaucase KV = 0 is the orig- inal context studied in string theory, it is possible to define invariants counting curves by describing curves by equations rather than in parametric form. Curves in V of genus g and class β ∈ H2(V ) given by equations are parametrized by Grothendieck’s Hilbert schemes Hilb(V ; β,χ) of subschemes of V with given Hilbert polynomial of degree 1. The invariants β, χ = 2 − g are encoded in the coefficients of the Hilbert polynomial. R. Thomas constructed a virtual fundamental class of Hilb(V ; β,χ) for three-folds V of dimension −β · KV , the same as the dimension of Mg,0(V, β). Thus one can define Donaldson–Thomas (DT) invariants as intersection numbers on this Hilbert scheme. There is no direct geometric relation between Hilb(V ; β,χ) and Mg,0(V, β), and indeed the (conjectural) relation between Gromov–Witten invariants and Donaldson–Thomas invariants is quite subtle. In its simplest form, it relates ∗ the GW invariants ¯ • ev γi to the DT invariants c (γi). The Mg,n(V,β) i Hilb(V ;β,χ) 2 ∗ class c2(γ ) is the coefficient of γ ∈ H (V ) in the Künneth decomposition of the second Chern class of the ideal sheaf of the universal family V ⊂ Hilb(V ; β,χ)× V . The conjecture of Maulik, Nekrasov, Okounkov and Pandharipande [6], [7], in- spired by ideas of string theory [14] states that suitably normalized generating func- tions ZGW (γ ; u)β , ZDT (γ ; q)β are essentially related by a coordinate transformation:
−d −d/2 iu (−iu) ZGW (γ1,...,γn; u)β = q ZDT (γ1,...,γn; q)β , if q =−e , β = 0.
Here d =−β · KV is the virtual dimension. Moreover these authors conjecture that there ZDT (γ ; q)β is a rational function of q. This has the important consequence that all (infinitely many) GW invariants are determined in principle by finitely many DT invariants. Versions of these conjectures are proven for local curves and the total space of the canonical bundle of a toric surface. The GW/DT correspondence can be viewed as a far-reaching generalization of formula (2), to which it reduces in the case where V is the product of a curve with C2.
4. Other uses of partitions
Here is a short account of other results of Okounkov based on the occurrence of partitions. One early result of Okounkov [9] is his first proof of the Baik–Deift–Johansson conjecture (two further different proofs followed, one by Borodin, Okounkov and Olshanski and one by Johansson). This conjecture states that, as n →∞, the joint distribution of the first few rows of a random partition of n with the Plancherel measure P (λ) = (dim λ)2/|λ|!, natural from representation theory, is the same, after proper shift and rescaling, as the distribution of the first few eigenvalues of a Gaussian random 60 Giovanni Felder hermitian matrix of size n. The proof involves comparing random surfaces given by Feynman diagrams and by ramified coverings and contains many ideas that anticipate Okounkov’s later work on Gromov–Witten invariants. Random partitions also play a key role in the work [8] of Nekrasov and Okounkov on N = 2 supersymmetric gauge theory in four dimensions. Seiberg and Witten gave a formula for the effective “prepotential”, postulating a duality with a theory of monopoles. The Seiberg–Witten formula is given in terms of periods on a family of algebraic curves, closely connected with classical integrable systems. Nekrasov showed how to rigorously define the prepotential of the gauge theory as a regularized instanton sum given by a localization integral on the moduli space of antiselfdual connections on R4. Nekrasov and Okounkov show that this localization integral can be written in terms of a measure on partitions with periodic potential and identify the Seiberg–Witten prepotential with the surface tension of the limit shape. Partitions of n also label (C×)2-invariant ideals of codimension n in C[x,y] and thus appear in localization integrals on the Hilbert scheme of points in the plane. Ok- ounkov and Pandharipande [13] describe the ring structure of the equivariant quantum cohomology (genus zero GW invariants) of this Hilbert scheme in terms of a time- dependent version of the Calogero–Moser operator from integrable systems.
5. Dimers
Dimers are a much studied classical subject in statistical mechanics and graph com- binatorics. Recent spectacular progress in this subject is due to the discovery by Okounkov and collaborators of a close connection of planar dimer models with real algebraic geometry. A dimer configuration (or perfect matching) on a bipartite graph G is a subset of the set of edges of G meeting every vertex exactly once. For example if G is a square grid we may visualize a dimer configuration as a tiling of a checkerboard by dominoes. In statistical mechanics one assigns positive weights (Boltzmann weights) to edges of G and defines the weight of a dimer configuration as the product of the weights of its edges. The basic tool is the Kasteleyn matrix of G, which is up to certain signs the weighted adjacency matrix of G. For finite G Kasteleyn proved that the partition function (i.e., the sum of the weights of all dimer configurations) is the absolute value of the determinant of the Kasteleyn matrix. Kenyon, Okounkov and Sheffield consider a doubly periodic bipartite graph G embedded in the plane with doubly periodic weights. For each natural number n one then has a probability measure on dimer configurations on Gn = G/nZ2 and statistical mechanics of dimers is essentially the study of the asymptotics of these probability measures in the thermodynamic limit n →∞. One key observation is the × 2 2 Kasteleyn matrix on G1 can be twisted by a character (z, w) ∈ (C ) of Z and thus one defines the spectral curve as the zero set P(z,w) = 0 of the determinant of the The work of Andrei Okounkov 61 twisted Kasteleyn matrix P(z,w) = det K(z,w). This determinant is a polynomial in z±1,w±1 with real coefficients and thus defines a real plane curve. The main observation of Kenyon, Okounkov and Sheffield [3] is that the spec- tral curve belongs to the very special class of (simple) Harnack curves, which were studied in the 19th century and have reappeared recently in real algebraic geome- try. Kenyon, Okounkov and Sheffield show that in the thermodynamic limit, three different phases (called gaseous, liquid and frozen) arise. These phases are character- ized by qualitatively different long-distance behaviour of pair correlation functions. One can see these phases by varying two real parameters (B1,B2) ( the “magnetic field”) in the weights, so that the spectral curve varies by rescaling the variables. The regions in the (B1,B2)-plane corresponding to different phases are described in terms of the amoeba of the spectral curve, namely the image of the curve by the map Log: (z, w) → (log |z|, log |w|). The amoeba of a curve is a closed subset of the plane which looks a bit like the microorganism with the same name. The amoeba itself corresponds to the liquid phase, the bounded components of its complement to the gaseous phase and the unbounded components to the frozen phase. This in- sight has a lot of consequences for the statistics of dimer models and lead Okounkov and collaborators to beautiful results on interfaces with various boundary conditions [3], [5]. Such a precise and complete description of phase diagrams and shapes of interfaces is unprecedented in statistical mechanics.
6. Random surfaces
One useful interpretation of dimers is as models for random surfaces in three-dimen- sional space. In the simplest case one considers a model for a melting or dissolving cubic crystal in which at a corner some atoms are missing (see figure).
A melting crystal corner (left) and the relation between tilings and dimers (right). Viewing the corner from the (1,1,1) direction one sees a tiling of the plane by 60 rhombi, which is the same as a dimer configuration on a honeycomb lattice (each tile 62 Giovanni Felder covers one dimer of the dimer configuration). In this simple model one gives the same probability for every configuration with given missing volume. If one lets the size of the cubes go to zero keeping the missing volume fixed, the probability measure concentrates on an a surface, the limit shape. More generally, every planar dimer model can be rephrased as a random surface model and limit shapes for more general crystal corner geometries can be defined. Kenyon, Okounkov and Sheffield show that the limit shape is given by the graph of (minus) the Ronkin function R(x,y) = ( πi)−2 (P (exz, eyw))dzdw/zw 2 |z|=|w|=1 log of the spectral curve (in the case of the honeycomb lattice with equal weights, P(z,w) = z + w + 1). This function is affine on the complement of the amoeba and strictly convex on the amoeba. So the connected components of the complement of the amoeba are the projections of the facets of the melting crystal. In addition to this surprising connection with real algebraic geometry, random surfaces of this type are essential in the GW/DT correspondence, see Section 3, as they arise in localization integrals for DT invariants of toric varieties.
7. The moduli space of Harnack curves
The notions used by Okounkov and collaborators in their study of dimer models arose in an independent recent development in real algebraic geometry. Their result bring a new probabilistic point of view in this classical subject. In real algebraic geometry, unsurmountable difficulties already appear when one consider curves. The basic open question is the first part of Hilbert’s 16th problem: what are the possible topological types of a smooth curve in the plane given by a polynomial equation P(z,w) = 0 of degree d? Topological types up to degree 7 are known but very few general results are available. In a recent development in real algebraic geometry in the context of toric varieties the class of Harnack curves plays an important role and can be characterized in many equivalent way. In one definition, due to Mikhalkin, a Harnack curve is a curve such that the map to its amoeba is 2:1 over the interior, except at possible nodal points; equivalently, by a theorem of Mikhalkin and Rullgård, a Harnack curve is a curve whose amoeba has area equal to the area of the Newton polygon of the polynomial P . These equivalent properties determine the topological type completely. Kenyon and Okounkov prove [4] that every Harnack curve is the spectral curve of some dimer model. They obtain an explicit parametrization of the moduli space of Harnack curves with fixed Newton polygon by weights of dimer models, and deduce in particular that the moduli spaces are connected. The work of Andrei Okounkov 63
8. Concluding remarks
Andrei Okounkov is a highly creative mathematician with both an exceptional breadth and a sense of unity of mathematics, allowing him to use and develop, with perfect ease, techniques and ideas from all branches of mathematics to reach his research objectives. His results not only settle important questions and open new avenues of research in several fields of mathematics, but they have the distinctive feature of mathematics of the very best quality: they give simple complete answers to important natural questions, they reveal hidden structures and new connections between math- ematical objects and they involve new ideas and techniques with wide applicability. Moreover, in addition to obtaining several results of this quality representing sig- nificant progress in different fields, Okounokov is able to create the ground, made of visions, intuitive ideas and techniques, where new mathematics appears. A striking example for this concerns the relation to physics: many important developments in mathematics of the last few decades have been inspired by high energy physics, whose intuition is based on notions often inaccessible to mathematics. Okounkov’s way of proceeding is to develop a mathematical intuition alternative to the intuition of high energy physics, allowing him and his collaborators to go beyond the mere verification of predictions of physicists. Thus, for example, in approaching the topological vertex of string theory, instead of stacks of D-branes and low energy effective actions we find mathematically more familiar notions such as localization and asymptotics of prob- ability measures. As a consequence, the scope of Okounkov’s research programme goes beyond the context suggested by physics: for example the Maulik–Nekrasov– Okounkov–Pandharipande conjecture is formulated (and proved in many cases) in a setting which is much more general than the Calabi–Yau case arising in string theory.
References
[1] Bloch, Spencer, and Okounkov, Andrei, The character of the infinite wedge representation. Adv. Math. 149 (1) (2000), 1–60. [2] Eskin, Alex, and Okounkov, Andrei, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145 (1) (2001), 59–103. [3] Kenyon, Richard, Okounkov, Andrei, and Sheffield, Scott, Dimers and amoebae. Ann. of Math. (2) 163 (3) (2006), 1019–1056. [4] Kenyon, Richard, and Okounkov, Andrei, Planar dimers and Harnack curves. Duke Math. J. 131 (3) (2006), 499–524. [5] Kenyon, Richard, and Okounkov, Andrei, Limit shapes and the complex burgers equation. Preprint; arXiv:math-ph/0507007. [6] Maulik, D., Nekrasov, N., Okounkov, A., and Pandharipande, R., Gromov-Witten theory and Donaldson-Thomas theory, I. Preprint; arXiv:math.AG/0312059. [7] Maulik, D., Nekrasov, N., Okounkov, A., and Pandharipande, R., Gromov-Witten theory and Donaldson-Thomas theory, II. Preprint; arXiv:math. AG/0406092. 64 Giovanni Felder
[8] Nekrasov, Nikita A., and Okounkov, Andrei, Seiberg-Witten theory and random partitions. In The unity of mathematics, Progr. Math. 244, Birkhäuser, Boston, MA, 2006, 525–596. [9] Okounkov, Andrei, Random matrices and random permutations. Internat. Math. Res. No- tices 2000 (20) (2000), 1043–1095. [10] Okounkov, Andrei, and Pandharipande, Rahul, Gromov-Witten theory, Hurwitz theory, and completed cycles. Ann. of Math. (2) 163 (2) (2006), 517–560. [11] Okounkov, Andrei, and Pandharipande, Rahul, The equivariant Gromov-Witten theory of P1. Ann. of Math. (2) 163 (2) (2006), 561–605. [12] Okounkov, Andrei, and Pandharipande, Rahul, Virasoro constraints for target curves. In- vent. Math. 163 (1) (2006), 47–108. [13] Okounkov,Andrei, and Pandharipande, Rahul, Quantum cohomology of the Hilbert scheme of points in the plane. Preprint; arXiv:math.AG/0411210. [14] Okounkov, Andrei, Reshetikhin, Nikolai, and Vafa, Cumrun, Quantum Calabi-Yau and classical crystals. In The unity of mathematics, Progr. Math. 244, Birkhäuser, Boston, MA, 2006, 597–618.
Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland E-mail: [email protected] 65
Grigory Perelman
Ph.D. Leningrad State University, November 1990
Positions until today December 1990–November 1992 Junior researcher at St. Petersburg Department of the V. A. Steklov Institute of the Russian Academy of Sciences (PDMI) December 1992–December 1998 Researcher at PDMI Computer Science Department January 1999–May 2004 Senior researcher at PDMI June 2004–December 2005 Leading researcher at PDMI The work of Grigory Perelman
John Lott
Grigory Perelman has been awarded the Fields Medal for his contributions to geom- etry and his revolutionary insights into the analytical and geometric structure of the Ricci flow. Perelman was born in 1966 and received his doctorate from St. Petersburg State University. He quickly became renowned for his work in Riemannian geometry and Alexandrov geometry, the latter being a form of Riemannian geometry for metric spaces. Some of Perelman’s results in Alexandrov geometry are summarized in his 1994 ICM talk [20]. We state one of his results in Riemannian geometry. In a short and striking article, Perelman proved the so-called Soul Conjecture.
Soul Conjecture (conjectured by Cheeger–Gromoll [2] in 1972, proved by Perelman [19] in 1994). Let M be a complete connected noncompact Riemannian manifold with nonnegative sectional curvatures. If there is a point where all of the sectional curvatures are positive then M is diffeomorphic to Euclidean space.
In the 1990s, Perelman shifted the focus of his research to the Ricci flow and its applications to the geometrization of three-dimensional manifolds. In three preprints [21], [22], [23] posted on the arXiv in 2002–2003, Perelman presented proofs of the Poincaré conjecture and the geometrization conjecture. The Poincaré conjecture dates back to 1904 [24]. The version stated by Poincaré is equivalent to the following.
Poincaré conjecture. A simply-connected closed (= compact boundaryless) smooth 3-dimensional manifold is diffeomorphic to the 3-sphere.
Thurston’s geometrization conjecture is a far-reaching generalization of the Poin- caré conjecture. It says that any closed orientable 3-dimensional manifold can be canonically cut along 2-spheres and 2-tori into “geometric pieces” [27]. There are various equivalent ways to state the conjecture. We give the version that is used in Perelman’s work.
Geometrization conjecture. If M is a connected closed orientable 3-dimensional manifold then there is a connected sum decomposition M = M1 # M2 # ··· # MN such that each Mi contains a 3-dimensional compact submanifold-with-boundary Gi ⊂ Mi with the following properties:
1. Gi is a graph manifold.
2. The boundary of Gi, if nonempty, consists of 2-tori that are incompressible in Mi.
Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2007 European Mathematical Society The work of Grigory Perelman 67
3. Mi −Gi admits a complete finite-volume Riemannian metric of constant negative curvature.
In the statement of the geometrization conjecture, Gi is allowed to be ∅ or Mi.(For 3 3 example, if M = S then we can take M1 = G1 = S .) The geometrization conjecture implies the Poincaré conjecture. Thurston proved that the geometrization conjecture holds for Haken 3-manifolds [27]. Background information on the Poincaré and geometrization conjectures is in [17]. Perelman’s papers have been scrutinized in various seminars around the world. At the time of this writing, the work is still being examined. Detailed expositions of Perelman’s work have appeared in [1], [16], [18].
1. The Ricci flow approach to geometrization
Perelman’s approach to the geometrization conjecture is along the lines of the Ricci flow strategy developed by Richard Hamilton. In order to put Perelman’s results in context, we give a brief summary of some of the earlier work. A 1995 survey of the field is in [12]. If M is a manifold and {g(t)} is a smooth one-parameter family of Riemannian metrics on M then the Ricci flow equation is dg =− . dt 2 Ric (1)
It describes the time evolution of the Riemannian metric. The right-hand side of the equation involves the Ricci tensor Ric of g(t). We will write (M, g(·)) for a Ricci flow solution. Ricci flow was introduced by Hamilton in 1982 in order to prove the following landmark theorem.
Positive Ricci curvature ([7]). Any connected closed 3-manifold M that admits a Riemannian metric of positive Ricci curvature also admits a Riemannian metric of constant positive sectional curvature.
A connected closed 3-dimensional Riemannian manifold with constant positive sectional curvature is isometric, up to scaling, to the quotient of the standard round 3-sphere by a finite group that acts freely and isometrically on S3. In particular, if M is simply-connected and admits a Riemannian metric of positive Ricci curvature then M is diffeomorphic to S3. The idea of the proof of the theorem is to run the Ricci flow, starting with the initial metric g(0) of positive Ricci curvature. The Ricci flow will go singular at some finite time T , caused by the shrinking of M to a point. As time approaches T , if one continually rescales M to have constant volume then the rescaled sectional curvatures become closer and closer to being constant on M. 68 John Lott
In the limit, one obtains a Riemannian metric on M with constant positive sectional curvature. Based on Hamilton’s result, Hamilton and S.-T.Yau developed a program to attack the Poincaré conjecture using Ricci flow. The basic idea was to put an arbitrary initial Riemannian metric on the closed 3-manifold, run the Ricci flow and analyze the evolution of the metric. Profound results about Ricci flow were obtained over the years by Hamilton and others. Among these results are the Hamilton–DeTurck work on the existence and uniqueness of Ricci flow solutions [6], [7], Hamilton’s maximum principle for Ricci flow solutions [8], the Hamilton–Chow analysis of Ricci flow on surfaces [4], [9], Shi’s local derivative estimates [25], Hamilton’s differential Harnack inequality for Ricci flow solutions with nonnegative curvature operator [10], Hamilton’s compact- ness theorem for Ricci flow solutions [11] and the Hamilton–Ivey curvature pinching estimate for three-dimensional Ricci flow solutions [12, Theorem 24.4],[15]. We state one more milestone result of Hamilton, from 1999. Nonsingular flows ([14]). Suppose that the normalized Ricci flow on a connected closed orientable 3-manifold M has a smooth solution that exists for all positive time and has uniformly bounded sectional curvatures. Then M satisfies the geometrization conjecture. The normalized Ricci flow is a variant of the Ricci flow in which the volume is kept constant. The above result clearly showed that Ricci flow was a promising approach to the geometrization conjecture. The remaining issues were to remove the assumption that the Ricci flow solution is smooth for all positive time, and the a priori bound on the sectional curvature. Regarding the smoothness issue, many 3-dimensional solutions of the Ricci flow equation (1) encounter a singularity within a finite time. One example of a singularity is a standard neckpinch, in which a cross-sectional 2-sphere {0}×S2 in a topological neck ((−1, 1) × S2) ⊂ M shrinks to a point in a finite time. Hamilton introduced the idea of performing a surgery on a neckpinch [13]. At some time, one removes a neighborhood (−c, c) × S2 of the shrinking 2-sphere and glues three-dimensional balls onto the ensuing boundary 2-spheres {−c}×S2 and {c}×S2. After the surgery operation the topology of the manifold has changed, but in a controllable way, since the presurgery manifold can be recovered from the postsurgery manifold by connected sums. One then lets the postsurgery manifold evolve by Ricci flow. If one encounters another neckpinch singularity then one performs a new surgery, lets the new manifold evolve, etc. One basic issue was to show that if the Ricci flow on a closed 3-manifold M encounters a singularity then an entire connected component disappears or there are nearby 2-spheres on which to do surgery. To attack this, Hamilton initiated a blowup analysis for Ricci flow [12, Section 16]. It is known that singularities arise from curvature blowups [7]. That is, if a Ricci flow solution exists on a maximal time interval [0,T), with T<∞, then limt→T − supx∈M | Riem(x, t)|=∞, where Riem The work of Grigory Perelman 69
{(x ,t )}∞ denotes the sectional curvatures. Suppose that i i i=1 is a sequence of spacetime points with limi→∞ | Riem(xi,ti)|=∞. In order to understand the geometry of the Ricci flow solution as one approaches the singularity time, one would like to 1 spatially expand around (xi,ti) by a factor of | Riem(xi,ti)| 2 and take a convergent subsequence of the ensuing geometries as i →∞. In fact, if one also expands the time coordinate by | Riem(xi,ti)| then one can consider taking a subsequence of rescaled Ricci flow solutions, that converges to a limit Ricci flow solution (M∞,g∞(·)). Hamilton’s compactness theorem [11] gives sufficient conditions to extract a con- vergent subsequence. Roughly speaking, on the rescaled solutions one needs uniform curvature bounds on balls and a uniform lower bound on the injectivity radius at (xi,ti). One can get the needed curvature bounds by carefully choosing the blowup points (xi,ti). However, before Perelman’s work, the needed injectivity radius bound was not available in full generality. If the blowup limit (M∞,g∞(·)) exists then it is a nonflat ancient solution, mean- ing that it is defined for t ∈ (−∞, 0]. The manifold M∞ may be compact or noncom- pact. In the three-dimensional case, Hamilton–Ivey pinching implies that for each t ∈ (−∞, 0], the time-t slice (M∞,g∞(t)) has nonnegative sectional curvature. Thus the possible blowup limits are very special. Hamilton gave detailed analyses of var- ious singularity models [12, Section 26]. One troublesome possibility, the so-called R × cigar soliton ancient solution, could not be excluded. If this particular solution occurred in a blowup limit then it would be problematic for the surgery program, as there would be no evident 2-spheres along which to do surgery. Hamilton conjectured [12, Section 26] that the R × cigar soliton solution could be excluded by means of a suitable generalization of the “little loop lemma” [12, Section 15]. In addition, there was the issue of showing that any point of high curvature in the original Ricci flow solution on [0,T)has a neighborhood that is indeed modeled by a blowup limit.
2. No local collapsing theorem
Perelman’s first breakthrough in Ricci flow, the no local collapsing theorem, removed two major stumbling blocks in the program to prove the geometrization of three- dimensional manifolds using Ricci flow. It allows one to take blowup limits of finite time singularities and it shows that the R × cigar soliton solution cannot arise as a blowup limit.
No local collapsing theorem ([21]). Let M be a closed n-dimensional manifold. If (M, g(·)) is a given Ricci flow solution that exists on a time interval [0,T), with T<∞, then for any ρ>0 there is a number κ>0 with the following property. Suppose that r ∈ (0,ρ) and let Bt (x, r) be a metric r-ball in a time-t slice. If the B (x, r) 1 sectional curvatures on t are bounded in absolute value by r2 then the volume n of Bt (x, r) is bounded below by κr . 70 John Lott
Perelman expresses the conclusion of the no local collapsing theorem by saying that the Ricci flow solution is “κ-noncollapsed at scales less than ρ”. The theorem says that after rescaling the metric ball to have radius one, if the sectional curvatures of the rescaled ball are bounded in absolute value by one then the volume of the rescaled ball is bounded below by κ. This lower bound on the volume is a form of noncollapsing. The theorem is scale-invariant, except for the condition that r should be less than the scale ρ. Perelman proves his no local collapsing theorem using new monotonic quantities for Ricci flows, which he calls the W-functional and the reduced volume V˜ . Expres- sions that are time-nondecreasing under the Ricci flow, loosely known as entropies, were known to be potentially useful tools; for example, such an entropy was used in the two-dimensional case in [9]. However, no relevant entropies were previously known in higher dimensions. Perelman’s entropy functionals arise from a new and profound understanding of the underlying structure of the Ricci flow equation. The method of proof of the no local collapsing theorem is to show that a local collapsing contradicts the monotonicity of the entropy. The significance of the no local collapsing theorem is that under a curvature as- sumption, it implies a lower bound on the injectivity radius at x, using [3]. This is what one needs in order to extract blowup limits. Any blowup limit (M∞,g∞(t)) will be a nonflat ancient solution which, from the no local collapsing theorem, is κ- noncollapsed at all scales. If such an ancient solution additionally has nonnegative cur- vature operator and bounded curvature (which will be the case for three-dimensional blowup limits) then Perelman calls it a κ-solution. Hereafter we assume that M is an orientable three-dimensional manifold. Perel- man gives the following classification of κ-solutions. Three-dimensional κ-solutions ([21], [22]). Any three-dimensional orientable κ- solution (M∞,g∞(·)) falls into one of the following types:
(a) (M∞,g∞(·)) is a finite isometric quotient of the round shrinking 3-sphere. (b) M∞ is diffeomorphic to S3 or RP 3. (M ,g (·)) R × S2 Z R × S2 (c) ∞ ∞ is the standard shrinking or its 2-quotient Z2 . (d) M∞ is diffeomorphic to R3 and, after rescaling, each time slice is asymptoti- cally necklike at infinity. In particular, Perelman shows that the R × cigar soliton ancient solution cannot arise as a blowup limit (as there is no κ>0 for which it is κ-noncollapsed at all scales), thereby realizing Hamilton’s conjecture. Perelman’s main use of κ-solutions is to model the high-curvature regions of a Ricci flow solution. By means of a sophisticated version of the blowup analysis, he proves the following result, which we state in a qualitative form. Canonical neighborhoods ([21]). Given T<∞,if(M, g(·)) is a nonsingular Ricci flow on a closed orientable 3-manifold M that is defined for t ∈[0,T)then any region The work of Grigory Perelman 71 of high scalar curvature is modeled, after rescaling, by the corresponding region in a three-dimensional κ-solution. Perelman’s first Ricci flow paper [21] concludes by showing that if the Ricci flow on a closed orientable 3-manifold M has a smooth solution that exists for all positive time then M satisfies the geometrization conjecture. There is no a priori curvature assumption. The proof of this result uses the long-time analysis described in Section 4.
3. Ricci flow with surgery
Perelman’s second Ricci flow paper [22] is a technical tour de force. He constructs a surgery algorithm to handle Ricci flow singularities. There are several issues involved in setting up a surgery algorithm. The most basic issue is to know that if the Ricci flow encounters a singularity then a connected component disappears or there are 2-spheres along which one can perform surgery, with control on the topology of the excised regions. This is an issue about the geometry of the Ricci flow near a singularity. For the first singularity time, it is handled by the above canonical neighborhood theorem. A second issue is to perform the surgery so as to not ruin the Hamilton–Ivey pinching condition on the curvature. A third issue is to show that surgery times do not accumulate. If surgery times accumulate then one may never get to a sufficiently large time to draw any topological conclusions. Hereafter we consider a Ricci flow (M, g(·)) on a connected closed oriented 3- manifold. With T being the first singularity time (if there is one), Perelman defines to be the points in M where the scalar curvature R stays bounded up to time T , i.e. M − ={x ∈ M : limt→T − R(x,t) =∞}. Here is an open subset of M. The next result gives the topology of M if the scalar curvature blows up everywhere. Components that go extinct ([22]). If =∅then M is diffeomorphic to a finite S3/ S1 × S2 S1 × S2 = RP 3 isometric quotient of the round 3-sphere, to or to Z2 # RP 3. Now suppose that the scalar curvature blows up somewhere but not everywhere, i.e. =∅. Perelman’s surgery procedure involves going up to the singularity time T and then trimming off horns. More precisely, there is a limiting time-T metric g on , with scalar curvature function R. For a small number ρ>0, the part of where the − scalar curvature is not too big is ρ ={x ∈ : R(x) ≤ ρ 2}, a compact subset of M. The connected components of can be divided into those that intersect ρ and those that do not. The connected components of that do not intersect ρ have uniformly large scalar curvature and are discarded. Using the canonical neighborhood theorem, Perelman shows that if a connected component of intersects ρ then it has a finite number of ends, each being a so-called “ε-horn”. The latter statement means that the scalar curvature goes to infinity as one exits the end, and in addition if x is a point in the ε-horn then after expanding the metric to make the scalar curvature 72 John Lott at x equal to one, there is a neighborhood of x in that is geometrically close to a cylinder (−ε−1,ε−1) × S2. (Here ε is a fixed small number.) The surgery procedure consists of cutting each such ε-horn along one of these cross-sectional 2-spheres and gluing in a 3-ball. If one does the surgery in this way then one has control on how the topology changes. Indeed, the presurgery manifold is recovered from the postsurgery manifold by taking connected sums of components, along with some possible additional con- nected sums with a finite number of S1 × S2 or RP 3 factors. (The S1 × S2 factors come from surgeries that do not disconnect M. The RP 3 factors can arise from con- nected components of that were thrown away.) One can guarantee that the surgery preserves the Hamilton–Ivey curvature pinching condition by carefully prescribing the geometric way that the 3-ball is glued, following [13]. One can then run the Ricci flow, starting from the postsurgery manifold, up to the next singularity time T (if there is one). However, if one wants to do surgery at time T then one must find the 2-spheres along which to cut. The main problem is that the earlier surgeries could invalidate the conclusion of the canonical neighborhood theorem on [0,T ). The proof of the canonical neighborhood theorem in turn relied on the no local collapsing theorem. One ingredient of Perelman’s resolution of this problem is to perform surgery sufficiently far down in the ε-horns. In effect, there is a self-improvement phenomenon as one goes down the horn. Perelman shows that for any δ>0, if a point x is in an ε-horn as before, and is sufficiently deep within the horn, then after rescaling to make the scalar curvature at x equal to one, there is a neighborhood of x that is geometrically close to a cylinder (−δ−1,δ−1) × S2. Hence one can ensure that the surgeries are done within cylinders that are very long relative to the cross-section. This turns out to be a key to extending the no local collapsing theorem to the case when there are intervening surgeries within the time interval. To summarize, Perelman proves the following technically difficult theorem. Surgery algorithm ([22]). The surgery parameters can be chosen so that there is a well-defined Ricci-flow-with-surgery. The statement means that the Ricci-flow-with-surgery exists for all time. (It is not excluded that at some finite time the remaining manifold becomes the empty set.) Perelman’s proof is quite intricate and uses an induction on the time interval. In addition, for a given induction step, i.e. on a given time interval, he uses a contradiction argument to show that the surgery parameters can be chosen so as to ensure that versions of the no local collapsing theorem and the canonical neighborhood theorem hold on the time interval, despite possible intervening surgeries. Volume considerations show that only a finite number of surgeries occur on a finite time interval; any surgery within the time interval removes a definite amount of volume, but there is only so much volume available for removal. The work of Grigory Perelman 73
4. Long-time behavior
Once one has the Ricci-flow-with-surgery, in order to obtain topological information about the original manifold one needs to analyze the long-time behavior. One special case is when there is a finite extinction time, i.e. the manifold in the Ricci-flow-with- surgery becomes the empty set at some finite time. Using his characterization of components that go extinct and analyzing the topology change caused by surgeries, Perelman gives the possible topology of a manifold whose Ricci flow has a finite extinction time. Finite extinction time ([22]). If a Ricci-flow-with-surgery starting from M has a finite extinction time then M is diffeomorphic to a connected sum of finite isometric quotients of the round S3 and copies of S1 × S2. In his third Ricci flow paper [23], Perelman goes further and uses minimal disk arguments to give a condition that ensures a finite extinction time; see also [5]. No aspherical factors ([23]). If the Kneser–Milnor prime decomposition of M does not have any aspherical factors then a Ricci-flow-with-surgery starting with any initial Riemannian metric on M has a finite extinction time. When put together, the above two steps give the topological possibilities for a connected closed orientable 3-manifold M whose prime decomposition does not have any aspherical factors. In particular, if M is simply-connected then the above two steps say that M is diffeomorphic to a connected sum of 3-spheres, and hence is diffeomorphic to the 3-sphere. In the general case when the Ricci-flow-with-surgery may not have a finite extinc- tion time, the goal is to show that as time goes on, one sees the desired decomposition of the geometrization conjecture. There could be an infinite number of total surgeries. At the time of this writing it is not known whether this actually happens. Perelman had the insight that one can draw topological conclusions nevertheless. Let Mt be a connected component of the time-t manifold. (If t is a surgery time then we consider the postsurgery manifold.) If Mt admitted any metric with nonnegative scalar curvature then it would be flat or the corresponding Ricci flow would have finite extinction time, in which case the topology is understood. So we can assume that Mt carries no metric with nonnegative scalar curvature. Consider hereafter the 1 metric g(t)ˆ = t g(t) on Mt . Perelman defines the “thick” part of Mt as follows. Given − x ∈ Mt , let the intrinsic scale ρ(x,t)be the radius ρ such that infB(x,ρ) Riem =−ρ 2, where Riem is the sectional curvature of g(t)ˆ . For any w>0, the w-thick part of + Mt is given by M (w, t) ={x ∈ Mt : vol(B(x, ρ(x, t))) > wρ(x, t)3}. Itisnot + + excluded that M (w, t) =∅or M (w, t) = Mt . By definition, one has a lower curvature bound on the ball B(x,ρ(x,t)). Perelman shows by a subtle argument that for large t,ifx is in the w-thick part M+(w, t) then B(x,ρ(x,t)) actually has an effective upper curvature bound. Adapting arguments from [13], he then shows that for any w>0, as time goes on, M+(w, t) approaches the 74 John Lott w-thick part of a hyperbolic manifold whose cuspidal tori, if any, are incompressible in Mt . On the other hand, if w is small and x is not in the w-thick part then the ball B(x,ρ(x,t)) has a lower curvature bound and a relatively small volume compared to ρ(x,t)3. From Perelman’s earlier work in collapsing theory, he knew that 3-manifolds which are locally volume collapsed, with respect to a lower curvature bound, are graph manifolds. Putting this together, Perelman is able to achieve the remarkable feat of realizing the hyperbolic/graph dichotomy, without making any a priori curvature assumptions. Hyperbolic pieces ([22]). Given the Ricci-flow-with-surgery, there are a finite col- {(H ,x )}k lection i i i=1 of complete pointed finite-volume Riemannian 3-manifolds of constant sectional curvature − 1 , a decreasing function α(t) tending to zero and a 4 f : k B x , 1 → M t family of maps t i=1 i α(t) t such that for large , 1. ft is α(t)-close to being an isometry. + 2. The image of ft contains M (α(t), t). f {H }k M 3. The image under t of a cuspidal torus of i i=1 is incompressible in t . That is, for larget, the α(t)-thick part of Mt is well approximated by the corre- k H M sponding subset of i=1 i. The remainder of t is highly collapsed with respect to a local lower curvature bound. Y k H Graph manifold pieces ([22], [26]). Let t be the truncation of i=1 i obtained 1 x by removing horoballs at distance approximately 2α(t) from the basepoints i. Then for large t, Mt − ft (Yt ) is a graph manifold. The above two steps, along with the fact that the components that go extinct are graph manifolds, and the fact that presurgery manifolds can be reconstructed from postsurgery manifolds via connected sums, imply the geometrization conjecture. Grigory Perelman has revolutionized the fields of geometry and topology. His work on Ricci flow is a spectacular achievement in geometric analysis. Perelman’s papers show profound originality and enormous technical skill. We will certainly be exploring Perelman’s ideas for many years to come.
References
[1] Cao, H.-D., and Zhu, X.-P.,A complete proof of the Poincaré and Geometrization Conjec- tures – application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10 (2006), 165–492. [2] Cheeger, J., and Gromoll, D., On the structure of complete manifolds of nonnegative curvature. Ann. of Math. 96 (1972), 413–443. [3] Cheeger, J., Gromov, M., and Taylor, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17 (1982), 15–53. The work of Grigory Perelman 75
[4] Chow, B., The Ricci flow on the 2-sphere. J. Differential Geom. 33 (1991), 325–334. [5] Colding, T., and Minicozzi, W., Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman. J. Amer. Math. Soc. 18 (2005), 561–569. [6] DeTurck, D., Deforming metrics in the direction of their Ricci tensors. J. Differential Geom. 18 (1983), 157–162. [7] Hamilton, R., Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982), 255–306. [8] Hamilton, R., Four-manifolds with positive curvature operator. J. Differential Geom. 24 (1986), 153–179. [9] Hamilton, R., The Ricci flow on surfaces. In Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math. 71, Amer. Math. Soc., Providence, RI, 1988, 237–262. [10] Hamilton, R., The Harnack estimate for the Ricci flow. J. Differential Geom. 37 (1993), 225–243. [11] Hamilton, R., A compactness property for solutions of the Ricci flow. Amer. J. Math. 117 (1995), 545–572. [12] Hamilton, R., The formation of singularities in the Ricci flow. In Surveys in differential geometry (Cambridge, MA, 1993), Vol. II, Internat. Press, Cambridge, MA, 1995, 7–136. [13] Hamilton, R., Four-manifolds with positive isotropic curvature. Comm. Anal. Geom. 5 (1997), 1–92. [14] Hamilton, R., Non-singular solutions of the Ricci flow on three-manifolds. Comm. Anal. Geom. 7 (1999), 695–729. [15] Ivey, T., Ricci solitons on compact three-manifolds. Differential Geom. Appl. 3 (1993), 301–307. [16] Kleiner, B., and Lott, J., Notes on Perelman’s papers. http://www.arxiv.org/abs/math. DG/0605667 (2006), based on earlier versions posted at http://www.math.lsa.umich. edu/research/ricciflow/perelman.html. [17] Milnor, J., Towards the Poincaré conjecture and the classification of 3-manifolds. Notices Amer. Math. Soc. 50 (2003), 1226–1233. [18] Morgan, J., and Tian, G., Ricci flow and the Poincaré conjecture. http://www.arxiv.org/ abs/math.DG/0607607 (2006). [19] Perelman, G., Proof of the soul conjecture of Cheeger and Gromoll. J. Differential Geom. 40 (1994), 209–212. [20] Perelman, G., Spaces with curvature bounded below. in Proceedings of the International Congress of Mathematicians (Zürich, 1994), Vol. 1, Birkhäuser, Basel 1995, 517–525. [21] Perelman, G., The entropy formula for the Ricci flow and its geometric applications. http://arxiv.org/abs/math.DG/0211159 (2002). [22] Perelman, G., Ricci flow with surgery on three-manifolds. http://www.arxiv.org/abs/ math.DG/0303109 (2003). [23] Perelman, G., Finite extinction time for the solutions to the Ricci flow on certain three- manifolds. http://www.arxiv.org/abs/math.DG/0307245 (2003). [24] Poincaré, H., Cinquième complément à l’analysis situs. Rend. Circ. Math. Palermo 18 (1904), 45–110. 76 John Lott
[25] Shi, W.-X., Deforming the metric on complete Riemannian manifolds. J. Differential Geom. 30 (1989), 223–301. [26] Shioya, T., and Yamaguchi, T., Volume collapsed three-manifolds with a lower curvature bound. Math. Ann. 333 (2005), 131–155. [27] Thurston, W., Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. 6 (1982), 357–381.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, U.S.A. E-mail: [email protected] 77
Terence Tao
B.Sc. Flinders University, December 1991 Ph.D. Princeton University, June 1996 Positions until today
1996–present University of California, Los Angeles (first as assistant professor, now full professor) 1999 and 2000 University of New South Wales 2001–2003 Clay Mathematical Institute 2001–2003 Australian National University The work of Terence Tao
Charles Fefferman
Mathematics at the highest level has several flavors. On seeing it, one might say:
(A) What amazing technical power!
(B) What a grand synthesis!
(C) How could anyone not have seen this before?
(D) Where on earth did this come from?
The work of Terence Tao encompasses all of the above. One cannot hope to capture its extraordinary range in a few pages. My goal here is simply to exhibit a few contributions by Tao and his collaborators, sufficient to produce all the reactions (A)...(D). I shall discuss the Kakeya problem, nonlinear Schrödinger equations and arithmetic progressions of primes. Let me start with a vignette from Tao’s work on the Kakeya problem, a beautiful and fundamental question at the intersection of geometry and combinatorics. I shall state the problem, comment briefly on its significance and history, and then single out my own personal favorite result, by Nets Katz and Tao. The original Kakeya problem was to determine the least possible area of a plane region inside which a needle of length 1 can be turned a full 360 degrees. Besicovitch and Pál showed that the area can be taken arbitrarily small. In its modern form, the Kakeya problem is to estimate the fractal dimension of a “Besicovitch set” E ⊂ Rn, i.e., a set containing line segments of length 1 in all directions. There are several relevant notions of “fractal dimension”. Here, let us use the Minkowski dimension, defined in terms of coverings of E by small balls of a fixed radius δ. The Minkowski dimension is the infimum of all β such that, for small δ, E can be covered by δ−β balls of radius δ. We want to prove that any Besicovitch set E ⊂ Rn has Minkowski dimension at least β(n), with β(n) as large as possible. (Perhaps β(n) = n.) Regarding the central importance of this problem, perhaps it is enough to say that it is intimately connected with the multiplier problem for Fourier transforms, and with the restriction of Fourier transforms to hypersurfaces; these in turn are closely connected with non-linear PDE via Strichartz estimates and their variants. There are also connections with other hard, interesting problems in combinatorics. Let me sketch some of the history of the problem over the last 30 years. The basic result of the 1970s is that β(2) = 2. (This is due to Davies, and is closely related
Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2007 European Mathematical Society The work of Terence Tao 79 to the early work of A. Córdoba. See [7], [8].) In the 1980s, Drury [9] showed that β(n) ≥ n+1 n ≥ 2 for 3. (See also Christ et al [4].) Then, about 1990, J. Bourgain and, shortly afterwards, T. Wolff discovered that Besicovitch sets of small fractal dimension have geometric structure (they contain “bouquets” and “hairbrushes”). During the 1990s, Bourgain also discovered a con- nection between the Kakeya problem and Gowers’ work on the Balog–Szemerédi theorem from combinatorics. These insights led to small, hard-won improvements in the value of β(n). The work looks deep and forbidding. See [1], [2], [24]. The connection with Gowers’ work arises in the following result. (We write #(S) for the number of elements of a set S.)
Deep Theorem (Bourgain, using ideas from Gowers’ improvement of Balog–Sze- merédi). Let A, B be subsets of an abelian group, and let G ⊂ A × B. Assume that #(A), #(B), and #{a + b : (a, b) ∈ G} are at most N. Then #{a − b : (a, b) ∈ G}≤ CN2−1/13, for a universal constant C.
The point is that one improves on the trivial bound N 2. From the Deep Theorem, one quickly obtains a result on β(n) by slicing the set E with three parallel hyperplanes H, H , H , with H halfway between H and H . Enter Nets Katz and Terence Tao, who proved the following result in 1999.
Little Lemma. Under the assumptions of the Deep Theorem, we have #{a − b : (a, b) ∈ G}≤CN2−1/6, for a universal constant C.
Note that the Little Lemma is strictly sharper than the DeepTheorem, Nevertheless, its proof takes only a few pages, and can be understood by a bright high-school student. After reading the proof, one has not the faintest clue where the idea came from (see (D)). β(n) ≥ 4n+3 The Little Lemma and its refinements led to the estimate 7 for the Kakeya problem, which at the time was the best result known for n>8. In high dimensions, the high-school accessible paper [18] went further than all the deep, forbidding work that came before it. Since then, there has been further progress, with Nets Katz, Izabella Łaba, and Terence Tao playing a leading rôle. The subject still looks deep and forbidding. In particular, regarding (A), let me refer the reader to the tour-de-force [17] by these authors. Unfortunately, the complete solution to the Kakeya problem still seems far away. Next, I shall discuss “interaction Morawetz estimates”. This simple idea, with profound consequences for PDE, was discovered by the “I-Team”: J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and Terence Tao. Let me start with the 3D non- linear Schrödinger equation (NLS):
p−1 i∂t u + xu =±|u| u, u(x, 0) = u0(x) given, (1) where u is a complex-valued function of (x, t) ∈ R3 × R, and p>1isgiven. 80 Charles Fefferman
This equation is important in physics and engineering. For instance, it describes the propagation of light in a fiber-optic cable. The behavior of solutions of (1) de- pends strongly on the ± sign and on the value of p. In particular, the minus sign is “focussing”, and we may expect solutions of (1) to develop singularities; while the plus sign is “defocussing”, and we expect solutions of (1) to spread out over large regions of space, as t −→ ± ∞ . In the defocussing case, the non-linear term in (1) should eventually become negligibly small, and the solution of (1) ought to behave like a solution of the (linear) free Schrödinger equation (i∂t + x)u = 0. From now on, we restrict attention to the defocussing case. 2 We note two obvious conserved quantities for (1): the “mass” R3 |u(x, t)| dx, and the energy, 1 2 1 p+1 E = |∇xu(x, t)| dx + |u(x, t)| dx. 2 p + 1 R3 R3 How can we prove that solutions of (1) spread out for large time? A fundamental tool is the Morawetz estimate. C. Morawetz first discovered this wonderful, simple idea for the non-linear Klein–Gordon equation. Let me describe it here for cubic 3D NLS, i.e., for equation (1) with p = 3. There, the Morawetz estimate asserts that T |u(x, t)|4 dx dt ≤ C (− )1/4 u( ·,t) 2 , sup x L2(R3) (2) |x| 0 ≤ t≤T 0 R3 for any T>0 and any solution of (1). The good news is that (2) instantly shows that u must eventually become small in any given bounded region of space. (If not, then the left-hand side of (2) grows linearly in T as T −→ ∞ , while the right-hand side of (2) remains bounded, thanks to conservation of mass and energy.) The bad news is that (2) does not rule out a scenario in which u(x, t) remains concentrated near a moving center x = x0(t). The trouble is that the weight function 1 |x| is concentrated near x = 0, whereas u( ·,t)may be concentrated somewhere else. The I-Team found an amazingly simple and straightforward way to overcome the bad news. Let me sketch the idea, starting with the classic proof of (2). To derive (2), we start with the quantity x M (t) = u(x,¯ t) · · ∇ u(x, t) dx. 0 Im |x| x (3) R3
On one hand, M0(t) is controlled by the right-hand side of (2), when 0 ≤ t ≤ T .On the other hand, a computation using (1) shows that d dx |u(x, t)|4 M (t) = π 2|u( ,t)|2 + |∇ u(x, t)|2 + dx, dt 0 4 0 2 |x| |x| (4) R3 R3 where ∇ denotes the angular part of the gradient. The work of Terence Tao 81
The Morawetz estimate (2) follows at once. The I-Team simply replaced M0(t) by a weighted average of translates, 2 M(t) = My(t) |u(y, t)| dy, R3 where x − y M (t) = u(x,¯ t) · · ∇ u(x, t) dx. y Im |x − y| x R3 This puts the greatest weight on those y ∈ R3 where u(y, t) lives – an eminently sensible idea. Starting with M(t) and proceeding more or less as in the proof of the Morawetz estimate, one obtains easily the Interaction Morawetz Estimate. T |u(x, t)|4 dx dt ≤ C u( ·, ) 2 · (− )1/4 u( ·,t) 2 . 0 L2(R3) sup x L2(R3) 0≤t ≤T 0 R3 (5) Again, the right-hand side is bounded for large T , thanks to conservation of mass and energy. This time however, the left-hand side grows linearly in T ,evenifour solution is concentrated in a moving ball. The I-Team has overcome the bad news. They made it look effortless. Why did no one think of it before? (See (C).) Observe that the right-hand side of (5) is much weaker than the energy; we need only half an x-derivative, as opposed to a full gradient. The original purpose of the interaction Morawetz estimate was to derive global existence for cubic defocussing 3D NLS in Sobolev spaces in which the energy may be infinite. That is a big achievement (see [6]), but I will not discuss it further here, except to point out that the proof involves additional ideas and formidable work. Instead, let me say a few words about the defocussing quintic 3D NLS, i.e., the case p = 5 of equation (1). This equation is particularly natural and deep, because it is critical for the energy. One knows that finite-energy initial data lead to solutions for a short time, and that small-energy initial data lead to global solutions. The challenge is to prove global existence for initial data with large, finite energy. To appreciate the difficulty of the problem, we have only to turn to the tour-de-force [3] by J. Bourgain, solving the problem in the radially symmetric case. The general case is an order of magnitude harder; a singularity can form only at the origin in the radial case, but it may form anywhere in the general case. The I-Team settled the general case using a version of the interaction Morawetz estimate for quintic NLS (with cutoffs, which unfortunately greatly complicate the analysis). This is natural, since in a sense one must overcome the same bad news as before. Their result [5] is as follows. 82 Charles Fefferman
Theorem. Take p = 5 in the defocussing case in (1). Then, for any finite-energy s initial data u0, there is a global solution u(x, t) of NLS. If u0 belongs to H with s s>1, then u( ·,t)also belongs to H for all t. Moreover, there exist solutions u± of the free Schrödinger equation, such that 2 |∇x(u(x, t) − u±(x, t))| dx
R3 tends to zero as t tends to ±∞. I will not try to describe their proof, except to say that they use an interaction Morawetz estimate with cutoffs, along with ideas from Bourgain [3], especially the “induction on energy”, as well as other ideas that I cannot begin to describe here. The details are highly formidable; see (A). We come now to Tao’s great joint paper ([16]; see also Green [15]) with Ben Green, in which they prove the following result. Here again, #(S) denotes the number of elements of a set S. Theorem GT. There exist arbitrarily long arithmetic progressions of primes. More precisely, given k ≥ 3, there exist constants c(k) > 0 and N0(k) ≥ 1, such that for any N>N0(k), we have #{k-term arithmetic progressions among the primes less c(k)N2 than N} > . (log N)k The lower bound here agrees in order of magnitude with a natural guess. (Green and Tao are currently working on a more precise result, with an optimal c(k).) To convey something of the range and depth of the ideas in the proof, let me start with the classic theorem of Szemerédi on sets of positive density. Here, ZN denotes the cyclic group of order N.
Theorem Sz 1. Given k and δ, we have for large enough N that any subset E ⊂ ZN with #(E)>δ· N contains an arithmetic progression of length k. Szemerédi’s theorem also gives a lower bound for the number of k-term progres- sions in E. (See [23].) It is convenient to speak of functions f rather than sets E. (One obtains Theorem Sz 1 from Theorem Sz 2 below, simply by taking f to be the indicator function of E.) Thus, Szemerédi’s theorem may be rephrased as follows.
Theorem Sz 2. Given k, δ, the following holds for large enough N. Let f : ZN → R, with 0 ≤ f(x)≤ 1 for all x, and with
(1) Avx∈ZN f(x)>δ. Then
(2) Avx,r∈ZN {f(x)· f(x+ r)...f(x+ (k − 1)r)} ≥ c(k,δ) > 0, where c(k,δ) depends only on k,δ (and not on N or f). The work of Terence Tao 83
(In (1), (2) and similar formulas, “Av” denotes the mean.) In Theorems Sz 1 and 2, δ stays fixed as N grows. If instead we could take δ ∼ 1/ log N, then the Green–Tao theorem would follow. However, such an im- provement of Theorems Sz 1, 2 seems utterly out of reach, and may be false. There are three very different proofs of Theorems Sz 1, 2; they are due to Szemerédi [21], Furstenberg [10], and Gowers [14]. Without doing justice to the remarkable ideas in these arguments, let me just say that Szemerédi used combinatorics, Furstenberg used ergodic theory, and Gowers used (non-linear) Fourier analysis. It is hard to see anything in common in these three proofs. In a sense, the Green–Tao paper synthesizes them all, by quoting Theorem Sz 2 and using ideas that go back to the proofs of Furstenberg and Gowers. See (B). Green and Tao prove a powerful extension of Theorem Sz 2, in which the hypoth- esis 0 ≤ f(x)≤ 1 is replaced by 0 ≤ f(x)≤ ν(x) for a suitable non-negative weight function ν(x). The function ν(x) is assumed to satisfy three conditions, which we describe crudely here.
• Avx∈ZN ν(x) = 1.
• We assume an upper bound on the quantity m Av t ν (λ (x) ) x=(x1,...,xt ) ∈ (ZN ) i i=1
t for certain affine functions λ1,...,λm : (ZN ) −→ ZN .
• For any h1,...,hm ∈ ZN , we assume that { } Avx∈ZN ν(x + h1)...ν(x+ hm) ≤ τm(hi − hj ), i = j
for a function τm : ZN −→ R that satisfies q Avh∈ZN (τm(h) ≤ C(m,q)
for any q.
Such a function ν(x) is called a “pseudo-random measure” by Green and Tao. Their extension of Szemerédi’s theorem is as follows.
Theorem GTS (Green–Tao–Szemerédi). Let k,δ > 0, suppose N is large enough and let ν be a pseudo-random measure. Let f : ZN −→ R, with 0 ≤ f(x) ≤ ν(x), and with Avx∈ZN f(x)≥ δ.
Then Avx,r∈ZN {f(x)· f(x+ r)...f(x+ (k − 1)r)} ≥ c(k,δ) > 0, where c(k,δ) depends on k,δ, but not on N or f . 84 Charles Fefferman
The point is that there are pseudo-random measures ν(x) that are large on sparse subsets of ZN (e.g., the primes up to N). We will return to this point. Let me say a few words about the proof of Theorem GTS, and then afterwards describe how it applies to the primes. It is in the proof of Theorem GTS that Szemerédi’s theorem is combined with ideas from Furstenberg and Gowers. Green and Tao break up the function f into a “uniform” and an “anti-uniform” part, f = fU + fU ⊥ .
They expand out Avx,r∈ZN {f(x)· f(x + r)...f(x + (k − 1)r)} into a sum of terms
(3) Avx,r∈ZN {f0(x) · f1(x + r)...fk−1(x + (k − 1)r)}, where each fi is either fU or fU ⊥ .
The terms (3) that contain any factor fU are o(1), thanks to ideas that go back to Gowers’ proof. This leaves us with the term (3) in which each fi is fU ⊥ . Let us call this the “critical term”. To control that term, Green and Tao partition ZN into subsets E1,E2,...,EA, ¯ and then define a function fU ⊥ on ZN by averaging fU ⊥ over each Eα. Green and Tao then prove that ¯ (4) replacing fU ⊥ by fU ⊥ makes a difference o(1) in the critical term, and moreover, ¯ ¯ (5) 0 ≤ fU ⊥ ≤ 1 and Avx∈ZN fU ⊥ (x) ≥ δ. ¯ Consequently, the classic Szemerédi theorem (Theorem Sz 2) applies to fU ⊥ , completing the proof of the Green–Tao–Szemerédi theorem. The proof of (4) and (5) is based on ideas that go back to Furstenberg’s proof of Szemerédi’s theorem. Once the Green–Tao–Szemeréditheorem is established, one can take f(x)= log x for x prime, f(x) = 0 otherwise. If we can find a pseudo-random measure ν such that (6) 0 ≤ f(x)≤ ν(x) for all x, then Theorem GTS applies, and it yields arbitrarily long arithmetic progressions of primes as in Theorem GT.A first guess for ν(x)is the standard Von Mangoldt function (x) = log p for x = pk, p prime; (x) = 0 otherwise. may indeed be a pseudo- random measure, but that would be very hard to prove. Fortunately, another function ν can be seen to be a pseudo-random measure satisfying (6), thanks to important work of Goldston–Yıldırım [11], [12], [13], using not-so-hard analytic number theory. Thus, in the end, a great theorem on the prime numbers is proven without hard analytic number theory. The difficulty lies elsewhere. The work of Terence Tao 85
I have repeatedly used the phrase “tour-de-force”; I promise that I am not exag- gerating. There are additional first-rate achievements by Tao that I have not mentioned at all. For instance, he has set forth a program [22] for proving the global existence and regularity of wave maps, by using the heat flow for harmonic maps. This has an excellent chance to work, and it may well have important applications in general relativity. I should also mention Tao’s joint work with Knutson [19] on the saturation conjecture in representation theory. It is most unusual for an analyst to solve an outstanding problem in algebra. Tao seems to be getting stronger year by year. It is hard to imagine what can top the work he has already done, but we await Tao’s future contributions with eager anticipation.
References
[1] Bourgain, J., Besicovitch-type maximal operators and applications to Fourier analysis. Geom. Funct. Anal. 1 (2) (1991), 147–187. [2] Bourgain, J., On the dimension of Kakeya sets and related maximal inequalities. Geom. Funct. Anal. 9 (2) (1999), 256–282. [3] Bourgain, J., Global well-posedness of defocusing 3D critical NLS in the radial case. J. Amer. Math. Soc. 12 (1999), 145–171. [4] Christ, M., Duoandikoetxea, J., Rubio de Francia, J. L., Maximal operators associated to the Radon transform and the Calderón-Zygmund method of rotations. Duke Math. J. 53 (1986), 189–209. [5] Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T., Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3. Ann. of Math.,to appear. [6] Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T., Global existence and scattering for rough solutions of a nonlinear Schrödinger equation in R3. Comm. Pure Appl. Math. 57 (8) (2004), 987–1014. [7] Córdoba,A., The Kakeya maximal function and the spherical summation multipliers. Amer. J. Math. 99 (1977), 1–22. [8] Davies, R., Some remarks on the Kakeya problem. Proc. Cambridge Philos. Soc. 69 (1971), 417–421. [9] Drury, S., Lp estimates for the x-ray transform. Illinois J. Math. 27 (1983), 125–129. [10] Furstenberg, H., Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Analyse Math. 31 (1977), 204–256. [11] Goldston, D., and Yıldırım, C. Y., Higher correlations of divisor sums related to primes, I: Triple correlations. Integers 3 (2003), A5 (electronic). [12] Goldston, D., and Yıldırım, C. Y., Higher correlations of divisor sums related to primes, III: k-correlations. Preprint, September 2002; arXiv: math.NT/0209102. [13] Goldston, D., and Yıldırım, C. Y., Small gaps between primes, I. Preprint, April 2005; arXiv:math.NT/0504336. 86 Charles Fefferman
[14] Gowers, T., A new proof of Szemerédi’s theorem. Geom. Funct. Anal 11 (2001), 465–588. [15] Green, B., Roth’s theorem in the primes. Ann. of Math. 161 (2005), 1609–1636. [16] Green, B., and Tao, T., The primes contain arbitrarily long arithmetic progressions. Ann. of Math., to appear. [17] Katz, N. H., Łaba, I., Tao, T., An improved bound on the Minkowski dimension of Besi- covitch sets in R3. Ann. of Math. 152 (2000), 383–446. [18] Katz, N. H., and Tao, T., Bounds on arithmetic projections, and applications to the Kakeya conjecture. Math. Res. Lett. 6 (1999), 625–630. [19] Knutson, A., and Tao, T., The honeycomb model of GLn(C) tensor products I: Proof of the saturation conjecture. J. Amer. Math. Soc. 12 (4) (1999), 1055–1090. [20] Morawetz, C., Time decay for the nonliner Klein-Gordon equation. Proc. Roy. Soc. Ser. A 306 (1968), 291–296. [21] Szemerédi, E., On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 (1975), 299–345. [22] Tao, T., Geometric renormalization of large energy wave maps. Preprint, November 2004; arXiv:math.AP/0411354. [23] Varnavides, P., On certain sets of positive density. J. London Math. Soc. 34 (1959), 358–360. [24] Wolff, T., Recent work connected with the Kakeya problem. In Prospects in mathematics (Princeton, NJ, 1996), Amer. Math. Soc., Providence, RI, 1999, 129–162.
Department of Mathematics, Fine Hall, WashingtonRoad, Princeton, NJ 08544-1000, U.S.A. E-mail: [email protected] 87
Wendelin Werner
Ph.D. Université Paris-Sud, 1993 Positions until today
1991–1993 Charge de recherche CNRS 1993–1995 European postdoctoral fellowship, University of Cambridge 1995–1997 Charge de recherche CNRS Since 1997 Professor at the Université Paris-Sud (Orsay) The work of Wendelin Werner
Charles M. Newman∗
1. Introduction
It is my great pleasure to briefly report on some of Wendelin Werner’s research ac- complishments that have led to his being awarded a Fields Medal at this International Congress of Mathematicians of 2006. There are a number of aspects of Werner’s work that add to my pleasure in this event. One is that he was trained as a probabilist, receiving his Ph.D. in 1993 under the supervision of Jean-François Le Gall in Paris with a dissertation concerning planar Brownian Motion – which, as we shall see, plays a major role in his later work as well. Until now, Probability Theory had not been represented among Fields Medals and so I am enormously pleased to be here to witness a change in that history. I myself was originally trained, not in Probability Theory, but in Mathematical Physics. Werner’s work, together with his collaborators such as Greg Lawler, Oded Schramm and Stas Smirnov, involves applications of Probablity and Conformal Map- ping Theory to fundamental issues in Statistical Physics, as we shall discuss. A second source of pleasure is my belief that this, together with other work of recent years, represents a watershed in the interaction between Mathematics and Physics generally. Namely, mathematicians such as Werner are not only providing rigorous proofs of already existing claims in the Physics literature, but beyond that are provid- ing quite new conceptual understanding of basic phenomena – in this case, a direct geometric picture of the intrinsically random structure of physical systems at their critical points (at least in two dimensions). One simple but important example is percolation – see Figure 1. Permit me a somewhat more personal remark as director of the Courant Insti- tute for the past four years. We have a scientific viewpoint, as did our predecessor institute in Göttingen – namely, that an important goal should be the elimination of ar- tificial distinctions between the Mathematical Sciences and their applications in other Sciences – I believe Wendelin Werner’s work brilliantly lives up to that philosophy. Yet a third source of pleasure concerns the collaborative nature of much ofWerner’s work. Beautiful and productive mathematics can be the result of many different personal workstyles. But the highly interactive style, of which Werner, together with Lawler, Schramm and his other collaborators, is a leading exemplar, appeals to many of us as simultaneously good for the soul while leading to work stronger than the sum of its parts. It is a promising sign to see Fields Medals awarded for this style of work.
∗ Research partially supported by the U.S. NSF under grants DMS-01-04278 and DMS-0606696.
Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2007 European Mathematical Society The work of Wendelin Werner 89
Figure 1. Finite portion of a (site) percolation configuration on the triangular lattice. Each hexagon represents a site and is assigned one of two colors. In the critical percolation model, colors are assigned randomly with equal probability. The cluster interfaces are indicated by heavy lines.
2. Brownian paths and intersection exponents
The area of Probability Theory which most strongly interacts with Statistical Physics is that involving stochastic processes with nontrivial spatial structure. This area, which also interacts with Finance, Communication Theory, Theoretical Computer Science and other fields, has long combined interesting applications with first-class Mathematics. Recent developments however have raised the perceived mathematical status of the best work from “merely” first-class to outstanding. Let me mention two pieces of Werner’s work from 1998–2000. These are not only of intrinsic significance, but also were precursors to the breakthroughs about to happen in the understanding of two-dimensional critical systems with (natural) conformal invariance. (There were of course other significant precursors, such as Aizenman’s path approach to scaling limits – see, e.g., [1], [2] – and Kenyon’s work on loop-erased walks and domino tilings – see, e.g., [13].) The first of these two pieces of work is a 1998 paper of Bálint Tóth and Werner [36]. The motivation was to construct a continuum version of Tóth’s earlier lattice “true self-repelling walk” and this led to a quite beautiful mathematical structure (an ex- tended version of a mostly unpublished and nearly forgotten construction done almost 20 years earlier by Arratia) of coalescing and “reflecting” one-dimensional Brownian paths, running forward and backward in time and filling up all of two-dimensional space-time. There is a (random) plane-filling curve within this structure that is anal- ogous to one that arises in scaling limits of uniformly random spanning trees and was 90 Charles M. Newman one of Schramm’s motivations in his 2000 paper introducing SLE [33]. SLE is an acronym for what was originally called the Stochastic Loewner Evolution and is now often called the Schramm–Loewner Evolution; more about SLE shortly. The second piece of work consists of two papers with Lawler in 1999 and 2000 involving planar Brownian intersection exponents [25], [26]. In the second of these, it was shown that the same set of exponents must occur providing only that certain locality and conformal invariance properties are valid. This was a key idea which, combined with the introduction of SLE for the analysis of two-dimensional critical phenomena, led to a remarkable series of three papers in 2001–2002 by Lawler, Schramm and Werner [17], [18], [20] which yielded a whole series of intersection exponents. For example, let W 1(t), W 2(t),... be independent planar Brownian motions start- ing from distinct points at t = 0. Then the probability that the random curve segments n −ζ +o( ) W 1([0,t]),...,W ([0,t]) are all disjoint is t n 1 as t →∞for some constant ζn.
Theorem 1 ([18]). The intersection exponents ζn, for n ≥ 2, are given by 4n2 − 1 ζn = . (1) 24 This formula had been conjectured earlier by Duplantier and Kwon [12] and de- rived later by Duplantier [11] in a nonrigorous calculation based on two-dimensional quantum gravity. Despite the simplicity of the formula, prior to the introduction of SLE-based methods, its derivation by conventional stochastic calculus techniques appeared to be quite out of reach.
3. Conformal probability theory
The period from 2001 until the present has seen an explosion of interest in and appli- cations of the SLE approach. To discuss this, we first give a very brief introduction to SLE; some good general references are [32], [38], [16]. For, say, a Jordan do- main D in the complex plane with distinct points a,b on its boundary ∂D, and κ a positive parameter, (chordal) SLE with parameter κ, denoted SLEκ , is a certain random continuous path (a curve, modulo monotonic reparametrization) in the clo- sure D, starting at a and ending at b. When κ ≤ 4, SLEκ is (with probability one) a simple path that only touches ∂D at a and b. Loewner, in work dating back to the 1920s [28], studied the evolution from a to b of nonrandom curves in terms of a real-valued “driving function.” By conformally mapping D to the upper half plane H and suitably reparametrizing, one obtains (for say κ ≤ 4) a simple curve γ(t)in H for t ∈ (0, ∞) and conformal mappings gt from H\γ([0,t]) to H with gt satisfying Loewner’s evolution equation, 2 ∂t gt (z) = , (2) gt (z) − U(t) The work of Wendelin Werner 91 with driving function U(t) = gt (γ (t)). SLEκ corresponds to the choice of U(t) as the random function B(κt) where B is standard one-dimensional Brownian motion. When κ>4, some modifications are necessary, but (2) remains valid – even for κ ≥ 8 when the curves become plane-filling. Now back to the SLE-based advances of the recent past. Many of these concern or were motivated by (nonrigorous) results in the Statistical Physics literature about two- dimensional critical phenomena. Critical points of physical systems typically happen at very specific values of physical parameters, such as where the vapor pressure curve in a liquid/gas system ends. Critical systems have many remarkable properties, such as random fluctuations that normally are observable only on microscopic scales manifesting themselves macroscopically. A related feature is that many quantities at or approaching the critical point have power law behavior, with the non-integer powers, known as critical exponents (as well as other macroscopic features, such as the scaling limits we will discuss later), believed to satisfy “universality”, i.e., microscopically distinct models in the same spatial dimension should have the same exponents at their respective critical points. Two-dimensional critical systems turn out to have an additional remarkable property, which is at the heart of both the SLE approach and its predecessors in the physics literature – that is conformal invariance on the macroscopic scale. As in the case of Brownian intersection exponents, many of the SLE-based results in two dimensions were rigorous proofs of exponent values that had been derived earlier by nonrigorous arguments – primarily those of what is known in the Physics literature as “Conformal Field Theory”, which dates back to the work of Polyakov and collaborators in the 1970s and 1980s [31], [4], [5] – see also [10], [30], [9]. Other results were brand new. I’ll discuss a few of these in more detail, but, as noted before, what is particularly exciting is that the SLE-based approach is not solely a rigorization of what already had existed in the physics literature but also a conceptually quite complementary approach to that of Conformal Field Theory. Werner in particular has emphasized the need to understand that complementary relationship; this has led, e.g., to a focus on the “restriction property”, as in his paper about the conformally invariant measure on self-avoiding loops [39]. That paper is one example of a burgeoning interest in extending the original SLE focus on random curves to the case of random loops, but still with conformal invariance properties, both in the specific case of percolation scaling limits [6], [7] and in the more general contexts of Brownian “loop soups” [27], [37] and Conformal Loop Ensembles as currently being studied by Scott Sheffield and Werner. Next are some more examples of the results obtained in the last six years or so.
4. Brownian frontier
Let W(t) be a planar Brownian motion. The complement in the plane of the curve segment W([0,t]) is a countable union of open sets, one of which is infinite; the 92 Charles M. Newman boundary of that infinite component is called the Brownian frontier. As a consequence of deep relations that planar Brownian motion and its intersection exponents have with SLE6 and its exponents (see [19], [23]), Lawler, Schramm and Werner obtained the following, proving a celebrated conjecture of Mandelbrot [29]. Theorem 2 ([21]). The Hausdorff dimension of the planar Brownian frontier is 4/3.
5. Loop-erased walks
A different set of results are stated somewhat informally in the next theorem. They concern loop-erased random walks and related random objects on lattices. Unlike the percolation case discussed next, these results about continuum scaling limits, in which the lattice scale shrinks to zero, are not restricted to a particular lattice. Theorem 3 ([24]). Let D be (say) a Jordan domain in the plane; then the scaling limits of loop-erased random walk, the uniformly random spanning tree and the related lattice-filling curve in D are, respectively, (radial) SLE2, a continuum “SLE2-based tree” and the plane-filling (chordal) SLE8. Scaling limits of lattice models are among the most interesting and often the most difficult results. To do them well requires the successful combination of concepts and techniques from three different areas: conformal geometry (as in the classical Löwner evolutions where the driving function is nonrandom), stochastic analysis (since for SLE the driving function is a Brownian motion), and the probability theory of lattice models (e.g., random walks or percolation or Ising models or ...). The work of Werner combines all three ingredients admirably well.
6. Percolation
Before closing, let me discuss one more example which demonstrates how these three areas can interact – scaling limits of two-dimensional critical percolation. The physics community knew (nonrigorously) the exponent values and even some geometric in- formation in the form of specific formulas for scaling limits of crossing probabilities between boundary segments of domains. These formulas were derived by Cardy [9] following Aizenman’s conjecture that they should be conformally invariant – see [15]. But there was little understanding of the scaling limit geometry of objects like cluster “interfaces” – see Figure 1. In [33], Schramm argued that the limit of one particular interface, the “exploration path,” should be SLE6. Smirnov, for the triangular lattice, then proved [34] that (A) the crossing probabilities do converge to the conformally invariant Cardy formulas, sketched an argument as to how that could lead to (B) convergence of the whole exploration path to SLE6 and argued further that one should be able to extend these The work of Wendelin Werner 93 results to (C) a “full scaling limit” for the family of “interface loops” of all clusters. In [35], Smirnov and Werner then proved certain percolation exponents, using explo- ration path convergence (B), while in [22], Lawler, Schramm and Werner combined the full scaling limit (C) with percolation arguments to prove another exponent value that is stated below. Convergence in (B) and (C) can be proved by using a considerable amount of lattice percolation machinery [6], [7], [8] – including results of Kesten, Sidoravicius and Zhang [14] about six-fold crossings of annuli and of Aizenman, Duplantier and Aharony [3] about narrow “fjords.” Then the percolation exponent results of Werner and coauthors apply and provide another excellent example of how the combination of the three ingredients mentioned above work together – e.g., the next theorem proves a prediction of den Nijs and Nienhuis [10], [30]. Theorem 4 ([22]). In critical site percolation on the triangular lattice, − / +o( ) Prob [cluster of origin has diameter ≥ R]=R 5 48 1 as R →∞. (3)
7. Conclusion
I close with some comments about continuum models of Probability Theory and their relation to other areas of Mathematics which are exemplified by the work of Wen- delin Werner. Traditionally, a major focus of Probability Theory, and especially so in France, has been on continuum objects such as Brownian Motion and Stochas- tic Calculus, with SLE and related processes as the latest continuum objects in the pantheon. Those of us raised in a different setting, such as Statistical Mechanics, sometimes regard lattice models as more “real” or “physical.” But this is a narrow view. It is only the continuum models which possess extra properties, like confor- mal invariance in the two-dimensional setting, that relate Probability Theory to other well-developed areas of Mathematics. Such relations and interactions have become of increasing importance in recent years and will continue to do so. Even if one is primarily interested in the original lattice models, it is quite clear that their properties, such as critical exponents and critical universality, cannot be understood without a deep analysis of the continuum models that arise in the scaling limit. Thanks to the work of Wendelin Werner, his collaborators, and others, one might say that now we are all “continuistas.”
References
[1] Aizenman, M., Burchard, A., Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 (1999), 419–453. [2] Aizenman, M., Burchard, A., Newman, C. M., Wilson, D., Scaling Limits for Minimal and Random Spanning Trees in Two Dimensions. Random Structures Algorithms 15 (1999), 319–367. 94 Charles M. Newman
[3] Aizenman, M., Duplantier, B., Aharony, A., Connectivity exponents and the external perimeter in 2D independent percolation. Phys. Rev. Lett. 83 (1999), 1359–1362. [4] Belavin, A. A., Polyakov, A. M., Zamolodchikov, A. B., Infinite conformal symmetry of critical fluctutations in two dimensions. J. Statist. Phys. 34 (1984), 763–774. [5] Belavin, A. A., Polyakov, A. M., Zamolodchikov, A. B., Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241 (1984), 333–380. [6] Camia, F., Newman, C. M., Continuum Nonsimple Loops and 2D Critical Percolation. J. Statist. Phys. 116 (2004), 157–173. [7] Camia, F., Newman, C. M.. Two-dimensional critical percolation: the full scaling limit. Commun. Math. Phys. 268 (2006), 1–38.
[8] Camia, F., Newman, C. M., Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Related Fields, submitted; arXiv:math.PR/0604487. [9] Cardy, J. L., Critical percolation in finite geometries. J. Phys. A 25 (1992), L201–L206. [10] den Nijs, M., A relation between the temperature exponents of the eight-vertex and the q-state Potts model. J. Phys. A 12 (1979), 1857–1868. [11] Duplantier, B., Random walks and quantum gravity in two dimensions. Phys. Rev. Lett. 81 (1998), 5489–5492. [12] Duplantier, B., Kwon, K.-H., Conformal invariance and intersection of random walks. Phys. Rev. Lett. 61 (1988), 2514–2517. [13] Kenyon, R., Conformal invariance of domino tiling. Ann. Probab. 28 (2000), 759–795. [14] Kesten, H., Sidoravicius, V., Zhang,Y., Almost all words are seen in critical site percolation on the triangular lattice. Electron. J. Probab. 3 (1998), paper no. 10. [15] Langlands, R., Pouliot, P., Saint-Aubin, Y., Conformal invariance for two-dimensional percolation. Bull. Amer. Math. Soc. 30 (1994), 1–61. [16] Lawler, G., Conformally Invariant Processes in the Plane. Math. Surveys Monogr. 114, Amer. Math. Soc., Providence, RI, 2005. [17] Lawler, G., Schramm, O., Werner, W., Values of Brownian intersection exponents I: Half- plane exponents. Acta Math. 187 (2001), 237–273. [18] Lawler, G., Schramm, O., Werner, W., Values of Brownian intersection exponents II: Plane exponents. Acta Math. 187 (2001), 275–308. [19] Lawler, G., Schramm, O., Werner, W., The dimension of the planar Brownian frontier is 4/3. Math. Res. Lett. 8 (2001), 401–411. [20] Lawler, G., Schramm, O., Werner, W.,Values of Brownian intersection exponents III: Two- sided exponents. Ann. Inst. Henri Poincaré 38 (2002), 109–123. [21] Lawler, G., Schramm, O., Werner, W., Analyticity of intersection exponents for planar Brownian motion. Acta Math. 189 (2002), 179–201. [22] Lawler, G., Schramm, O., Werner, W., One-arm exponent for critical 2D percolation. Electron. J. Probab. 7 (2002), paper no. 2, 1–13. [23] Lawler, G., Schramm, O., Werner, W., Conformal restriction: the chordal case. J. Amer. Math. Soc. 16 (2003), 917–955. [24] Lawler, G., Schramm, O., Werner, W., Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (2004), 939–995. The work of Wendelin Werner 95
[25] Lawler, G., Werner, W., Intersection exponents for planar Brownian motion. Ann. Probab. 27 (1999), 1601–1642. [26] Lawler, G., Werner, W., Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. 2 (2000), 291–328. [27] Lawler, G., Werner, W., The Brownian loop-soup. Probab. Theory Related Fields 128 (2004), 565–588. [28] Löwner, K. (Loewner, C.), Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I. Math. Ann. 89 (1923), 103–121. [29] B. B. Mandelbrot, The Fractal Geometry of Nature. Freeman, San Francisco, Calif., 1982. [30] Nienhuis, B., Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Statist. Phys. 34 (1984), 731–761. [31] Polyakov, A. M., Conformal symmetry of critical fluctuations. JETP Letters 12 (1970), 381–383. [32] Rohde, S., Schramm, O., Basic properties of SLE. Ann. Math. 161 (2005), 883–924. [33] Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221–288. [34] Smirnov, S., Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris 333 (2001), 239–244. [35] Smirnov, S., Werner, W., Critical exponents for two-dimensional percolation. Math. Rev. Lett. 8 (2001), 729–744. [36] Tóth, B., Werner, W., The true self-repelling motion. Probab. Theory Related Fields 111 (1998), 375–452. [37] Werner, W., SLEs as boundaries of clusters of Brownian loops. C. R. Acad. Sci. Paris 337 (2003), 481–486. [38] Werner, W., Random planar curves and Schramm-Loewner evolutions. In Lectures on probability theory and statistics, Lecture Notes in Math. 1840, Springer-Verlag, Berlin 2004, 107–195. [39] Werner, W., The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc., to appear; arXiv:math.PR/0511605.
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A. E-mail: [email protected] 96
Jon Kleinberg
A.B. in Mathematics and Computer Science, Cornell University, 1993 Ph.D. in Computer Science, MIT, 1996 Positions until today 1996–1997 Visiting Scientist, IBM Almaden Research Center 1996–present Faculty member, Cornell University Computer Science Department 2004–2005 Visiting Faculty, Carnegie Mellon University The work of Jon Kleinberg
John Hopcroft
Introduction
Jon Kleinberg’s research has helped lay the theoretical foundations for the information age. He has developed the theory that underlies search engines, collaborative filtering, organizing and extracting information from sources such as the World Wide Web, news streams, and the large data collections that are becoming available in astronomy, bioinformatics and many other areas. The following is a brief overview of five of his major contributions.
Hubs and authorities
In the 1960s library science developed the vector space model for representing doc- uments [13]. The vector space model is constructed by sorting all words in the vocabulary of some corpus of documents and forming a vector space model where each dimension corresponds to one word of the vocabulary. A document is repre- sented as a vector where the value of each coordinate is the number of times the word associated with that dimension appears in the document. Two documents are likely to be on a related topic if the angle between their vector representations is small. Early search engines relied on the vector space model to find web pages that were close to a query. Jon’s work on hubs and authorities recognized that the link structure of the web provided additional information to aid in tasks such as search. His work on hubs and authorities addressed the problem of how, out of the millions of documents on the World Wide Web (WWW), you can select a small number in response to a query. Prior to 1997 search engines selected documents based on the vector space model or a variant of it. Jon’s work on hubs and authorities [5] laid the foundation to rank pages based on links as opposed to word content. He introduced the notion of an authority as an important page on a topic and a hub as a page that has links to many authorities. Mathematically, an authority is a page pointed to by many hubs and a hub is a page that points to many authorities. For the concepts of hubs and authorities to be useful, one needs to develop the mathematics to identify hubs and authorities; that is, to break the cycle in the definition. The World Wide Web can be represented as a directed graph where nodes corre- spond to web pages and directed edges represent links from one page to another. Let A be the adjacency matrix for the underlying web graph. Jon did a text based search
Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2007 European Mathematical Society 98 John Hopcroft to find, say, the 200 most relevant web pages for a query based on word content. This set might not contain the most important web sites since, as he points out, the words “search engine” did not appear on the sites of the popular search engines in 1997, such as Alta Vista or Excite. Similarly there were over a million web sites containing the word “Harvard” but the site www.harvard.edu was not the site that contained the term “Harvard” most often. Thus, he expanded the set by adding web sites reached by either in or out links from the 200 sites. To avoid adding thousands of additional web sites when one of the web pages was extremely popular, he restricted each page in the set to add at most fifty additional pages to the original set. In the process of adding web pages, he ignored links to pages within the same domain name as these tended to be navigational links such as “top of page”. In the resulting sub graph, which was now likely to contain most of the important relevant pages, he assigned weights to pages and then iteratively adjusted the weights. Actually, each page was assigned two weights, a hub weight and an authority weight. The hub weights were updated by replacing the weight of each hub with the sum of the weights of the authorities that it points to. Next the weights of the authorities were updated by replacing the weight of each authority with the sum of the hub weights pointing to it. The hub weights and the authority weights were then normalized so that the sum of the squares of each set of weights equaled one. This iterative technique converges so that the hub weights are the coordinates of the major eigenvector of AAT and the authority weights are the coordinates of the major eigenvector of AT A. Thus, the eigenvectors of AAT and AT A rank the pages as hubs and authorities. This work allowed a global analysis of the full WWW link structure to be replaced by a much more local method of analysis on a small focused sub graph. This work is closely related to the work of Brin and Page [2] that lead to Google. Brin and Page did a random walk on the underlying graph of the WWW and computed the stationary probability of the walk. Since the directed graph has some nodes with no out degree, they had to resolve the problem of losing probability when a walk reached a node with no out going edges. Actually, they had to solve the more general problem of the probability ending up on sub graphs with no out going edges, leaving the other nodes with zero probability. The way this was resolved was that at each step the walk would jump with some small probability ε to a node selected uniformly at random and with probability 1−ε take a step of the random walk to an adjacent node. Kleinberg’s research on hubs and authorities has influenced the way that all major search engines rank pages today. It has also spawned an industry creating ways to help organizations get their web pages to the top of lists produced by search engines to various queries. Today there is a broad field of research in universities based on this work. The work of Jon Kleinberg 99
Small worlds
We are all familiar with the notion of “six degrees of separation”, the notion that any two people in the world are connected by a short string of acquaintances. Stanley Milgram [12] in the sixties carried out experiments in which a letter would be given to an individual in a state such as Nebraska with instructions to get the letter to an individual in Massachusetts by mailing it to a friend known on a first name basis. The friend would be given the same instructions. The length of the path from Nebraska to Massachusetts would typically be between five and six steps. Milgram’s experiments on this inter-personal connectivity lead to a substantial research effort in the social sciences focused on the interconnections in social net- works. From 1967 to 1999 this work was primarily concerned with the structure of relationships and the existence of short paths in social networks. Although the fact that individuals have the ability to actually find the short paths as was demonstrated by Milgram’s original experiment, there was no work on understanding how individuals actually found the short paths or what conditions were necessary for them to do so. In 1998 Watts and Strogatz [14] refined the concept of a small world, giving precise definitions and simple models. Their work captured the intuitive notion of a reference frame, such as the geographical location where people live or their occupation. In this reference frame, an individual is more likely to know the neighbor next door than a person in a different state. Most people know their neighbors but they also know some people who are far removed. The relationships between individuals and their neighbors were referred to as short links, and the few friends or relatives far away that the individuals knew were referred to as long-range links. One simple model developed by Watts and Strogatz was a circular ring of nodes where each node was connected to its nearest neighbors clockwise and counterclock- wise around the circle, as well as to a few randomly selected nodes that were far away. Watts and Strogatz proved that any pair of nodes is, with high probability, connected by a short path, thus justifying the terminology “small world”. Jon [6] raised the issue of how you find these short paths in a social network without creating a map of the entire social world. That is, how do you find a path using only local information? He assumed a rectangular grid with nodes connected to their four nearest neighbors, along with one random long range connection from each node. As the distance increased the probability of a long range random link connecting two nodes decreased. Jon’s model captured the concept of a reference frame with different scales of resolution: neighbor, same block, same city, or same country. Jon showed that when the decrease in probability was quadratic with distance, then there exists an efficient (polynomial time) algorithm for finding a short path. If the probability decreases slower or faster, he proved the surprising result that no efficient algorithm, using only local information, could exist for finding a short path even though a short path may exist. In Jon’s model the probability of a long range edge between nodes x and y de- creased as dist(x, y)−r where dist(x, y) is the grid distance between nodes x and y. 100 John Hopcroft
For r = 0, the probability of a long range contact is independent of distance. In this case, the average length of such a contact is fairly long, but the long range contacts are independent of the geometry of the grid and there is no effective way to use them in finding short paths even though short paths exist. As r increases, the average length of the random long range contacts decreases but their structure starts to become useful in finding short paths. At r = 2 these two phenomena are balanced and one can find short paths efficiently using only local knowledge. For r>2 the average length of the random long-range contact continues to decrease. Although short paths may still exist, there is no polynomial time algorithm using only local information for finding them. When r equals infinity, no long-range contacts exist and hence no short paths. What is surprising is that for r<2 or for r>2, no efficient algorithm using only local information exists for finding short paths. Theorem 1. Let G be a random graph consisting of an n × n grid plus an additional edge from each vertex u to some random vertex v where the probability of the edge (u, v) is inversely proportional to dist(u, v)r . Here dist(u, v) is the grid distance between vertices u and v.Forr = 2, there is a decentralized algorithm so that the expected time to find a path from some start vertex s to a destination vertex t is O(log2 n). Proof. At each step the algorithm selects the edge from its current location that gets it closest to its destination. The algorithm is said to be in phase j when the lattice distance from the current vertex to the destination t is in the interval (2j , 2j+1). Thus, there are at most log n phases. We will now prove that the expected time the algorithm remains in each phase is at most log n steps and, hence, the time to find a path is O(log2 n). For a fixed vertex u the probability that the long-distance edge from u goes to v is d(u, v)−2 . −2 w =u d(u, w) We wish to get an upper bound on the denominator so as to get a lower bound on the probability of an edge of distance d(u, v). Since the set of vertices at distance i from u forms a diamond centered at u with sides of length i, there are 4i vertices at distance i from vertex u, unless u is close to a boundary in which case there are fewer. Thus 2n−2 2n−2 − 1 1 d(u, w) 2 ≤ 4i = 4 . i2 i w =u i=1 i=1
For large n there exists a constant c1 such that −2 d(u, w) ≤ c1 ln n. w =u
It follows that there exists a constant c2 such that each vertex that is within distance j u c 2−2j u 2 of has probability of at least 2 ln n of being the long distance contact of . The work of Jon Kleinberg 101
The current step ends phase j of the algorithm if the vertex reached is within distance 2j of the destination t. In the plane, the number of vertices at distance i from a given vertex grows linearly with i. Thus, the number of vertices within distance 2j of the destination t is at least j 2 j j 2 (2 + 2) j− i = > 22 2. 2 i=1 Since the current location is within distance 2j+1 of t and since there are at least 22j−1 vertices within distance 2j of t, there are at least 22j−1 vertices within distance 2j+1 + 2j < 2j+2 of the current location. Each of these vertices that are within c distance 2j+2 of the current location has probability of at least 2 of being the ln(n)22j+4 long-distance contact.
> j+1 u 2 At least 2j−1 t vertices 2j 2j+1
If one of the 22j−1 vertices that are within distance 2j of t and within distance 2j+2 of the current location is the long-distance contact of u, it will be u’s closest neighbor to t. Thus, phase j ends with probability at least c 2j−1 c 22 = 2 . ln(n)22j+4 8ln(n) We now bound by log n the total time spent in step j.Forj ≤ log log n the current vertex is distance at most log n from the destination t. Thus, even taking only local edges suffices. For j>log log n, let xj be the number of steps spent in phase j. Then ∞ E(xj ) = i Prob(xj = i). i=1 Since
1 Prob(xi = 1) + 2 Prob(xi = 2) +···=Prob(xi ≥ 1) + Prob(xi ≥ 2) +··· we get ∞ ∞ i− c2 1 1 8lnn E(xj ) = Prob(xj ≥ i) ≤ 1 − = c = . 8ln(n) 1 − 1 − 2 c i=1 i=1 8ln(n) 2 Thus, the total number of steps is O(log2 n). 2 102 John Hopcroft
Even more surprising than the above result that states there exists an efficient local algorithm for finding short paths when the exponent r equals two, were Jon’s additional results proving no such algorithms exist when the exponent r was either greater than two or less than two. This research on finding paths in small worlds has found applications outside the social sciences in such areas as peer-to-peer file sharing systems. It turns out that many real sets of data have the needed quadratic decrease in probability distribution. For example, an on-line community where you measure distance between individuals by the distance between their zip codes, often has this distribution after distances are corrected for the highly nonuniform population density of the U.S. [11].
Bursts
In order to understand a stream of information, one may organize it by topic, time, or some other parameter. In many data streams a topic suddenly appears with high frequency and then dies out. The burst of activity provides a structure that can be used to identify information in the data stream. Jon’s work [7] on bursts developed the mathematics to organize a data stream by bursts of activity. If one is watching a news stream and the word Katrina suddenly appears, even if one does not understand English, one recognizes that an event has taken place somewhere in the world. The question is how do you automatically detect the sudden increase in frequency of a word and distinguish the increase from a statistical fluctuation? Jon developed a model in which bursts can be efficiently detected in a statistically meaningful manner. A simple model for generating a sequence of events is to randomly generate the events according to a distribution where the gap x between events satisfies the distribution p(x) = αe−αx. Thus, the arrival rate of events is α and the expected 1 value of the gap between events is α . A more sophisticated model has a set of states and state transitions. Associated with each state is an event arrival rate. In Jon’s model there is an infinite number of states q0, q1, …, each having an event 1 arrival rate. State q0 is the base state and has the base event rate g . Each state qi has a 1 i rate αi = g s where s is a scaling parameter. In state qi there are two transitions, one to the state qi+1 with higher event rate and one to qi−1 with lower event rate. There is a cost associated with each transition to a higher event rate state. Given a sequence of events, one finds the state sequence that most closely matches the gaps with the smallest number of state transitions. Jon applied the methodology to several data streams demonstrating that his method- ology could robustly and efficiently identify bursts and thereby provide a technique to organize the underlying content of the data streams. The data streams consisted of his own email, the papers that appeared in the professional conferences, FOCS and STOC, and finally the U.S. State of the Union Addresses from 1790 to 2002. The burst analysis of Jon’s email indicated bursts in traffic when conference or proposal The work of Jon Kleinberg 103 deadlines neared. The burst analysis of words in papers in the FOCS and STOC conferences demonstrated that the technique finds words that suddenly increased in frequency rather then finding words of high frequency over time. Most of the words indicate the emergence or sudden increase in the importance of a technical area, al- though some of the bursts correspond to word usage, such as the word “how” which appeared in a number of titles in the 1982 to 1988 period. The burst analysis of the U.S. State of the Union Addresses covered a 200 year time period from 1790 to 2002 and considered each word. Adjusting the scale parameter s produced short bursts of 5–10 years or longer bursts covering several decades. The bursts corresponded to national events up through 1970 at which time the frequency of bursts increased dramatically.
Table 1. Bursts in word usage in U.S. State of the Union Addresses. energy 1812–1814 rebellion 1861–1871 Korea 1951–1954 bank 1833–1836 veterans 1925–1931 Vietnam 1951–1954 California 1848–1852 wartime 1941–1947 inflation 1971–1980 slavery 1857–1860 atomic 1947–1959 oil 1974–1981
This work on bursts demonstrated that one could use the temporal structure of data streams, such as email, click streams, or search engine queries, to organize the material as well as its content. Organizing data streams around the bursts which occur, provides us with another tool for organizing material in the information age.
Nearest neighbor
Many problems in information retrieval and clustering involve the nearest neighbor problem in a high dimensional space. A good survey of important work in this area can be found in [3]. An important algorithmic question is how to preprocess n points in d- dimensions so that given a query vector, one can find its closest neighbor. An important version of this problem is the ε-approximation nearest neighbor problem. Given a set P of vectors in d-dimensional space and a query vector x, the ε-approximation nearest neighbor problem is to find a vector y in P such that for any z in P dist(x, y) ≤ (1 + ε) dist(x, z). Prior to Jon’s work on nearest neighbor search in high dimensions [4], there was much research on this problem. Early work asked how one could preprocess P so as to be able to efficiently find the nearest neighbor to a query vector x. Most of the previous papers required query time exponential in the dimension d. Thus, if the dimension of the space was larger than log n, there was no method faster than the brute force algorithm that uses time O(dn). Jon developed an algorithm for the ε-approximation nearest neighbor problem that improved on the brute force algorithm 104 John Hopcroft for all values of d. Jon’s work lead to an O((d log2 d)(d + log n)) time algorithm for the problem [4]. The basic idea is to project the set of points P onto random lines through the origin. If a point x is closer than a point y to the query q, then with probability greater 1 x y than 2 , the projection of will be closer than the projection of to the projection of the query q. Thus, with a sufficient number of projections, the probability that x is closer to the query than y in a majority of the projections, will be true with high probability. If (1 + ε) dist(x, q) ≤ dist(y, q), then the test will fail for a majority of projections only if a majority of the lines onto which P is projected come from an exceptional set. Using a VC-dimension argument, Jon showed that the probability of more than half the lines lying in an exceptional set is vanishingly small.
Collaborative filtering
An important problem in the information age is to target a response to a person based on a small amount of information. For example, if a customer orders an item from a network store, the store may want to send him or her an advertisement based on that order. Similarly, a search engine may want to target an ad to a customer based on a query. In the case of a purchase, if for every potential item one knew the probability that the customer would buy the item, they might target an ad for the item of highest probability. However, in the case where there are possibly hundreds of thousands of items, how does one learn the probability of a customer purchasing each item based on the purchase of two or three items? If the only structure of the problem is the matrix of probabilities of customers and items, there is probably little one could do. However, if the items fall into a small number of categories and the mechanism with which a customer buys an item is that he or she first chooses a category and then having chosen a category chooses an item, one could use the structure to help in estimating the probabilities of purchasing the various items. Suppose the customer/item probability matrix is the product of a customer/category matrix times a category/item matrix. Then one can acquire information about the category/item matrix from purchases of all customers, not just the purchases of one customer. Let A be the probability matrix of customers versus items. Then A = PW where P is the probability matrix of customers versus categories and W is the probability matrix of items given a specific category. Note that the rank of A is at most the number of categories.
item category item customer A = customer P = category W The work of Jon Kleinberg 105
Suppose we know W the matrix of probabilities of items given the categories. Let u bearowofA, the vector of probabilities with which a customer selects items. Let u˜ be an estimate of u obtained from s samples. That is, the ith component of the 1 vector u˜ is s times the number of times the customer selected the ith item out of s selections. The question is, how close will u˜ be to u? Observe that u is in the range of W and that u˜ most likely is not. Thus, projecting u˜ onto the range of W might improve the approximation. The question is what projection should be used? The obvious projection is to project orthogonally but this is not the only possibility. Recall that we know W. Let W be a generalized pseudo inverse of W.Foru in the range of W (a linear combination of the columns of W) WWu = u. However, for x not in the range of W, WWx is obviously not x but some vector in the range of W.
x
W −1
W W −1x
W −1
W −1u u
W
range W
Applying WW to u˜ − u we get WW(u˜ − u) = WWu˜ − u.
We need to bound how far the projection WWu˜ can be from u. What we would like is for each component of WWu˜ to be within ε of the corresponding component of u with high probability. Then, recommending the item corresponding to the largest component would be the optimal recommendation. If the maximum element of WW is B, then how large can any component of WWu˜ −u be? Stated another way, how large can v(u˜ −u) be for any vector v where every element of v is bounded by some constant B? Write s u˜ = 1 u¯ s i i=1 u¯ i vT u˜ = 1 s vT u¯ where i is the indicator vector for the th selection. Then s i=1 i. The terms in the summation are independent random variables since the selections are 106 John Hopcroft
v u¯ independent. The variance of the product of an element of times an element of i B 2 T B2 is at most s , and hence the variance of v u˜ is at most s . (Elements of u¯ i have 1 value 0 or s and the maximum of v is B.) Hence, by Chebyshev’s inequality, the probability that vT u˜ will differ from its expected value by more than ε is bounded.
2 T T B Prob(|v u˜ − v u| < . ε2s
The above result tells us that vT u˜ will be close to its expected value of vT u provided no element of v is excessively large. Thus, in projecting u˜ onto the space of W we want to use a projection WW where W is selected so that it has no excessively large element. This led Jon and his colleague Mark Sandler [8] to use linear programming to 1 find a pseudo inverse in which the maximum element was bounded by where = Wx minx1=1 1. This lead to a collaborative filtering algorithm that recommends an item whose probability of being purchased by the customer is within an ε of the highest probability item with high probability. What is so important about this work is that it can be viewed as the start of a theory based on the 1-norm. Although much of the theory of approximation is based on the 2-norm and in fact approximating a matrix A by a low rank matrix Ak one can prove that the Frobenius norm of the error matrix is minimized by techniques based on the 2-norm, the error is not uniformly distributed. Furthermore, the error is strongly influenced by outliers. Use of the 1-norm is a promising approach to these problems.
Closing remarks
This brief summary covers five important research thrusts that are representative of Kleinberg’s work. His web page contains many other exciting results of which I will mention three. First is his early work with Eva Tardos [9] on network routing and the disjoint paths problem. They developed a constant-factor approximation algorithm for the maximum disjoint paths problem in the two-dimensional grid graph. Given a designated set of terminal node pairs, one wants to connect as many pairs as possible by paths that are disjoint. Their algorithm extends to a larger class of graphs that generalizes the grid. Second is his early work with Borodin, Ragahavan, Sudan and Williamson [1] on the worst-case analysis of packet-routing networks. This work presents a framework for analyzing the stability of packet-routing networks in a worst-case model without probabilistic assumptions. Here one assumes packets are injected into the network, limited only by simple deterministic rate bounds, and then shows that certain standard protocols guarantee queues remain bounded forever while other standard protocols do not. The work of Jon Kleinberg 107
Third is his work with Eva Tardos on classification with pairwise relationships [10]. This paper gives an algorithm for classification in the following setting: We are given a set of objects (e.g. web pages) to classify, each into one of k different types, and we have both local information about each object, as well as link information specifying that certain pairs of objects are likely to have similar types. For example, in classifying web pages by topic (or into some other categories), one may have an estimate for each page in isolation, and also know that pairs of pages joined by hyperlinks are more likely to be about similar topics.
Conclusions
Jon’s work has laid a foundation for the science base necessary to support the infor- mation age. Not only is the work foundational mathematically but it has contributed to the economic growth of industries.
References
[1] Borodin,A., Kleinberg, J., Raghavan, P., Sudan, M., andWilliamson, D.,Adversarial queue- ing theory. In Proceedings of the 28nd Annual Symposium on Theory of Computing, 1996, 376–385. [2] Brin, S., and Page, L., Anatomy of a Large-Scale Hypertextual Web Search Engine. In Proceedings of the 7th International World Wide Web Conference, 1998, 107–117. [3] Indyk, P., Nearest Neighbors in High-dimensional Spaces. In Handbook of Discrete and Computational Geometry (J. E. Goodman and J. O’Rourke, eds.), 2nd edition, CRC Press, Boca Raton, FL, 2004, 877–892. [4] Kleinberg, J., Two algorithms for nearest-neighbor search in high dimensions. In Proceed- ings of the 29th ACM Symposium on Theory of Computing, ACM Press, New York 1997, 599–608. [5] Kleinberg, J., Authoritative sources in a hyperlinked environment. In Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, ACM, New York 1998, 668–677. [6] Kleinberg, J., The small-world phenomenon: An algorithmic perspective. In Proceedings of the 32nd Annual Symposium on Theory of Computing, ACM Press, New York 2000, 163–170. [7] Kleinberg, J., Bursty and Hierarchical Structure in Streams. In Proceedings of the 8th ACM SIGKDD Intl. Conf. on Knowledge Discovery and Data Mining, ACM, New York 2002, 91–101. [8] Kleinberg, J., and Sandler, M., Using Mixture Models for Collaborative Filtering. In Pro- ceedings of the 36th ACM Symposium on Theory of Computing, ACM, New York 2004, 569–578. [9] Kleinberg, J., and Tardos, E., Disjoint paths in densely embedded graphs. In Proceedings of the 36th IEEE Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 1995, 52–59. 108 John Hopcroft
[10] Kleinberg, J., and Tardos, E., Approximation Algorithms for Classification Problems with Pairwise Relationships: Metric Labeling and Markov Random Fields. In Proceedings of 40th IEEE Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 1999, 14–23. [11] Liben-Nowell, D., Novak, J., Kumar, R., Raghavan, P., andTomkins,A., Geographic routing in social networks. Proc. Natl. Acad. Sci. USA 102 (2005), 11623–11628. [12] Milgram, S., The small world problem. Psychology Today 1 (1967), 60–67. [13] Salton, G., Automatic Text Processing. Addison-Wesley, Reading, Mass., 1989. [14] Watts, D., and Strogatz, S., Collective dynamics of small-world networks. Nature 393 (1998), 440–442.
Department of Computer Science, Cornell University, Ithaca, New York 14953, U.S.A. On Kiyosi Itô’s work and its impact
Hans Föllmer
About a week before the start of the International Congress, an anonymous participant in a weblog discussion of potential candidates for the Fields medals voiced his concern that there might be a bias against applied mathematics and went on to write: “Iam hoping that the Gauss prize will correct this obvious problem and they will pick someone really wonderful like Kiyosi Itô of Itô Calculus fame”. Indeed this has happened: The Gauss Prize 2006 for Applications of Mathematics has been awarded to Kiyosi Itô “for laying the foundations of the theory of stochas- tic differential equations and stochastic analysis”. However, in his message to the Congress Kiyosi Itô says that he considers himself a pure mathematician, and while he was delighted to receive this honor, he was also surprised to be awarded a prize for applications of mathematics. So why is the Gauss prize so appropriate in his case, and why was this anonymous discussant who obviously cares about applied mathematics so enthusiastic? The statutes of the Gauss prize say that it is “to be awarded for • outstanding mathematical contributions that have found significant applica- tions outside of mathematics, or • achievements that made the application of mathematical methods to areas out- side of mathematics possible in an innovative way”. My aim is to show why, on both accounts, Kiyosi Itô is such a natural choice. Kiyosi Itô was born in 1915. The following photo was taken in 1942 when he was working in the Statistical Bureau of the Japanese Government:
Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2007 European Mathematical Society 110 Hans Föllmer
At this time he had just achieved a major breakthrough in the theory of Markov processes. The results first appeared in 1942 in a mimeographed paper “Differential equations determining a Markov process” written in Japanese (Zenkoku Sizyo Sugaku Danwakai-si). English versions and further extensions of these initial results were published between 1944 and 1951 in Japan; see [24]. These papers laid the foundations of the field which later became known as stochastic analysis. A systematic account appeared in the Memoirs of the American Mathematical Society in 1951 under the title “On stochastic differential equations” [23], thanks to J. L. Doob who immediately recognized the importance of Itô’s work. What was the breakthrough all about? A Markov process is usually described in terms of the transition probabilities Pt (x, A) which specify, for each state x and any time t ≥ 0, the probability of finding the process at time t in some subset A of the state space, given that x is the initial state at time 0. These transition probabilities should satisfy the Chapman–Kolmogorov equations Pt+s(x, A) = Pt (x, dy)Ps(y, A).
For the purpose of this exposition we limit the discussion to the special case of a dif- fusion process with state space Rd . A fundamental extension theorem of Kolmogorov guarantees, for each initial state x, the existence of a probability measure Px on the space of continuous paths d = C([0, ∞), R ) such that the conditional probabilities governing future positions are given by the transition probabilities, i.e.,
Px[Xt+s ∈ A|Ft ]=Ps(Xt , A).
Here we use the notation Xt (ω) = ω(t) for ω ∈ , and Ft denotes the σ-field generated by the path behavior up to time t. In analytical terms, the infinitesimal structure of the Markov process is described by the infinitesimal generator
Pt − I L := lim . (1) t↓0 t In the diffusion case, this operator takes the form
d d 1 ∂2 ∂ L = aij (x) + bi(x) (2) 2 ∂xi∂xj ∂xi i,j=1 i=1 with a state-dependent diffusion matrix a = (aij ) and a state-dependent drift vector b = (bi), and for any smooth function f the function u defined by u(x, t) := Pt f(x) satisfies Kolmogorov’s backward equation d ∂t u = Lu on R × (0, ∞). (3) On Kiyosi Itô’s work and its impact 111
Itô’s aim was to reach a deeper understanding of the dynamics by describing the infinitesimal structure of the process in probabilistic terms. His basic idea was to i) identify the “tangents” of the process, and to ii) (re-) construct the process pathwise from its tangents. At the level of stochastic processes, the role of “straight lines” is taken by processes whose increments are independent and identically distributed over time intervals of the same length. Such processes are named in honor of Paul Lévy. Kiyosi Itô had already investigated in depth the pathwise behavior of Lévy processes by proving what is now known as the Lévy–Itô decomposition [21]. In the continuous case and in dimension d = 1, the prototype of such a Lévy process is a Brownian motion with constant drift, whose increments have a Gaussian distribution with mean and variance proportional to the length of the time interval. This process had been introduced in 1900 by Louis Bachelier as a model for the price fluctuation on the Paris stock market, five years before Albert Einstein used the same model in connection with the heat equation. A standard Brownian motion, which starts in 0 and whose increments have zero mean and variance equal to the length of the time interval, is also named in honor of Norbert Wiener who in 1923 gave the first rigorous construction, and the corresponding measure on the space of continuous paths is usually called Wiener measure. An explicit construction of a Wiener process with time interval [0, 1] can be obtained as follows: Take a sequence of independent Gaussian random variables Y1,Y2,... with mean 0 and variance 1, defined on some probability space ( , F ,P), 2 and some orthonormal basis (ϕn)n=1,2,... in L [0, 1]. Then the random series ∞ t Wt (ω) = Yn(ω) ϕn(s) ds n=1 0 is uniformly convergent and thus defines a continuous curve, P -almost surely. Wiener had studied the special case of a trigonometric basis, and Lévy had simplified the com- putations by using the Haar functions. But the definitive proof that the construction works in full generality was given by Itô and Nisio [32] in 1968. In the case of a diffusion it is therefore natural to say that a “tangent” of the Markov process in a state x should be an affine function of the Wiener process with coefficients depending on that state. Thus Itô was led to describe the infinitesimal behavior of the diffusion by a “stochastic differential equation” of the form
dXt = σ(Xt )dWt + b(Xt )dt. (4) In d dimensions, the Wiener process is of the form W = (W 1,...,Wd ) with d in- dependent standard Brownian motions, and σ(x)is a matrix such that σ(x)σT (x) = a(x). The second part of the program now consisted in solving the stochastic differ- ential equation, i.e., constructing the trajectories of the Markov process in the form t t Xt (ω) = x + σ(Xs(ω)) dWs(ω) + b(Xs(ω)) ds. (5) 0 0 112 Hans Föllmer
At this point a major difficulty arose. Wiener et al. had shown that the typical path of a Wiener process is continuous but nowhere differentiable. In particular, a Brownian path is not of bounded variation and thus cannot be used as an integrator in the Lebesgue–Stieltjes sense. In order to make sense out of equation (5) it was thus necessary to introduce what is now known as the theory of “stochastic integration”. In their introduction to the Selected Papers [24] of Kiyosi Itô, D. Stroock and S. R. S. Varadhan write: “Everyone who is likely to pick up this book has at least heard that there is a subject called the theory of stochastic integration and that K. Itô is the Lebesgue of this branch of integration theory (Paley and Wiener were its Riemann)”. Wiener and Paley had in fact made a first step, using integration by parts to define the integral t t ξs dWs := ξt Wt − Ws dξs 0 0 for deterministic integrands of bounded variation, and then using isometry to pass to deterministic integrands in L2[0,t]. But this “Wiener integral” is no help for the problem at hand, since the integrand ξt = σ(Xt ) is neither deterministic nor of bounded variation. In a decisive step, Itô succeeded in giving a construction of much wider scope. Roughly speaking, he showed that the stochastic integral t ξ dW ≈ ξ (W − W ) s s ti ti+1 ti (6) 0 i can be defined as a limit of non-anticipating Riemann sums for a wide class of stochas- tic integrands ξ = (ξt ). These sums are non-anticipating in two ways. First, the integrand is evaluated at the beginning of each time interval. Secondly, the values ξt only depend on the past observations of the Brownian path up to time t and not on its future behavior. To carry out the construction, Kiyosi Itô used the isometry t 2 t 2 E ξs dWs = E ξs ds . 0 0 This is clearly satisfied for simple non-anticipating integrands which are piecewise constant along a fixed partition of the time axis. The appropriate class of general integrands and the corresponding stochastic integrals are obtained by taking L2-limits on both sides. In particular the Itô integral has zero expectation, since this property obviously holds for the non-anticipating Riemann sums in (6). Once Kiyosi Itô had introduced the stochastic integral in this way, it was clear how to define a solution of the stochastic differential equation in rigorous terms. In order to prove the existence of the solution, Itô used a stochastic version of the method of successive approximation, having first clarified the dynamic properties of stochastic integrals viewed as stochastic processes with time parameter t. In order to complete his program, Itô had to verify that his solution of the stochas- tic differential equation indeed yields a pathwise construction of the given Markov On Kiyosi Itô’s work and its impact 113 process. To do so, Itô invented a new calculus for smooth functions observed along the highly non-smooth paths of a diffusion. In particular he proved what is now known as Itô’s formula. In fact there are nowadays many practioners who may not know or may not care about Lebesgue and Riemann, but who do know and do care about Itô’s formula. In 1987 Kiyosi Itô received the Wolf Prize in Mathematics. The laudatio states that “he has given us a full understanding of the infinitesimal development of Markov sample paths. This may be viewed as Newton’s law in the stochastic realm, providing a direct translation between the governing partial differential equation and the un- derlying probabilistic mechanism. Its main ingredient is the differential and integral calculus of functions of Brownian motion. The resulting theory is a cornerstone of modern probability, both pure and applied”. The reference to Newton stresses the fundamental character of Itô’s contribution to the theory of Markov processes. Let us also mention Leibniz in order to emphasize the fundamental importance of Itô’s work from another point of view. In fact Itô’s approach can be seen as a natural extension of Leibniz’s algorithmic formulation of the differential calculus. In a manuscript written in 1675 Leibniz argues that the whole differential calculus can be developed out of the basic product rule d(XY) = XdY + YdX, (7) and he writes: “Quod theorema sane memorabile omnibus curvis commune est”.In particular, this implies the rule dX2 = 2XdX and, more generally,
df (X) = f (X)dX (8) for a smooth function f observed along the curve X. Since the 19th century we know, of course, that these rules are not “common to all (continuous) curves”, since a continuous curve does not have to be differentiable. But it was Kiyosi Itô who discovered how these rules can be modified in such a way that they generate a highly efficient calculus for the non-differentiable trajectories of a diffusion process. In Itô’s calculus, the classical rule dX2 = 2XdX is replaced by
dX2 = 2XdX + dX , where 2 X t = lim (Xt − Xt ) (9) n i+1 i ti ∈Dn ti He then went on to show that the behavior of a function f ∈ C2 observed along the paths of the solution is described by the rule 1 df (X) = f (X)dX + f (X) dX , (11) 2 which is now known as Itô’s formula. Note that a continuous curve of bounded variation has quadratic variation 0, and so Itô’s formula may indeed be viewed as an extension of the classical differentiation rule (8). More generally, the classical product rule (7) becomes a special case of Itô’s product rule d(XY) = XdY + YdX+ dX, Y , where X, Y denotes the quadratic covariation of X and Y , defined in analogy to (9) or, equivalently, by polarization: X, Y =1(X + Y −X −Y ). 2 For a smooth function f on Rd ×[0, ∞) and a continuous curve X = (X1,...,Xd ) such that the quadratic covariations Xi,Xj exist, the d-dimensional version of Itô’s formula takes the form d 1 i j df (X, t) =∇xf(X,t)dX+ ft (X,t)dt + fx x (X,t)dX ,X . (12) 2 i j i,j=1 Let us now come back to the original task of identifying the solution of the stochastic differential equation (4) as a pathwise construction of the original Markov process. In a first step, Itô showed that the solution is indeed a Markov process. Moreover he proved that the solution has quadratic covariations of the form t i j X ,X t = σi,k(Xs)σj,k(Xs)ds. 0 k Thus Itô’s formula for a smooth function observed along the paths of the solution reduces to ∂ df (X, t) =∇ f(X,t)σ(X)dW + (L + )f (X, t) dt, x ∂t (13) where L is given by (2). In order to show that L is indeed the infinitesimal generator of the Markovian solution process, it is now enough to take a smooth function on Rd and to use Itô’s formula in order to write t t Ex[f(Xt ) − f(X0)]=Ex ∇xf(Xs)σ (Xs)dWs + Lf(Xs)ds . 0 0 On Kiyosi Itô’s work and its impact 115 Recalling that the Itô integral appearing on the right-hand side has zero expectation, dividing by t and passing to the limit, we see that the infinitesimal generator associated to the transition probabilities of the Markovian solution process as in (1) coincides with the partial differential operator L defined by (2). With a similar application of Itô’s formula, Kiyosi Itô also showed that the solution of the stochastic differential equation satisfies Kolmogorov’s backward equation (3). This concludes our sketch of Itô’s construction of Markov processes as solutions of a corresponding stochastic differential equation. Let us emphasize, however, that we have outlined the argument only in the special case of a time-homogeneous diffusion process. In fact, Kiyosi Itô himself succeeded immediately in solving the problem in full generality, including time-inhomogeneous Markov processes with jumps and making full use of his previous analysis of general Lévy processes. For a comprehen- sive view of the general picture we refer to D. Stroock’s book Markov Processes from K. Itô’s Perspective [46] and, of course, to Kiyosi Itô’s original publications [24]. At this point let us make a brief digression to mention a parallel approach to the construction of diffusion processes which was discovered by Wolfgang Doeblin. Born in Berlin in 1915, son of the prominent Jewish writer Alfred Döblin who took his family into exile in 1933, he studied mathematics in Paris and published results on Markov chains which became famous in the fifties. It was much less known, however, that he had also worked on the probabilistic foundation of Kolmogorov’s equation. In February 1940, while serving in the French army and shortly before he took his life rather than surrender himself to the German troops, Wolfgang Doeblin sent a manuscript to the Academy of Sciences in Paris as a pli cacheté. This sealed envelope was finally opened in May 2000. The manuscript contains a representation of the paths of the diffusion process where the stochastic integral on the right hand side of equation (5) is replaced by a time change of Brownian motion. While Doeblin’s approach does not involve the theory of stochastic integration which was developed by Kiyosi Itô and which is crucial for the applications described below, it does provide an alternative solution to the pathwise construction problem, and it anticipates important developments in martingale theory related to the idea of a random time change; see Bru and Yor [4] for a detailed account of the human and scientific aspects of this startling discovery. Over the last 50 years the impact of Itô’s breakthrough has been immense, both within mathematics and over a wide range of applications in other areas. Within mathematics, this process took some time to gain momentum, at least in the West. On receiving Itô’s manuscript On stochastic differential equations, J. L. Doob immedi- ately recognized its importance and made sure that it was published in the Memoirs of the AMS in 1951. Moreover, in his book on Stochastic processes [9] which appeared in 1953, Doob devoted a whole chapter to Itô’s construction of stochastic integrals and showed that it carries over without any major change from Brownian motion to general martingales. But when Kiyosi Itô came to Princeton in 1954, at that time a stronghold of probability theory with William Feller as the central figure, his new ap- proach to diffusion theory did not attract much attention. Feller was mainly interested 116 Hans Föllmer in the general structure of one-dimensional diffusions with local generator d d L = , dm ds motivated by his intuition that a “one-dimensional diffusion traveler makes a trip in accordance with the road map indicated by the scale function s and with the speed indicated by the measure m”; see [30]. Together with Henry McKean, at that time a graduate student of Feller, Kiyosi Itô started to work on a probabilistic construction of these general diffusions in terms of Lévy’s local time. This program was carried out in complete generality in their joint book Diffusion Processes and Their Sample Paths [31], a major landmark in the development of probability theory in the sixties. At that time I was a graduate student at the University of Erlangen, and when a group of us organized an informal seminar on the book of Itô and McKean we found it very hard to read. But then we were delighted to discover that Itô’s own Lectures on Stochastic Processes [25] given at the Tata Institute were much more accessible; see also [26] and [27]. This impression was fully confirmed when Professor Itô came to Erlangen in the summer of 1968: We thoroughly enjoyed the stimulating style of his lectures as illustrated by the following photo (even though it was taken ten years later at Cornell University), and also his gentle and encouraging way of talking to the graduate students. Ironically, however, neither stochastic integrals nor stochastic differential equations were mentioned anywhere in the book, in the Tata lecture notes, or in his talks in Erlangen. The situation began to change in the sixties, first in the East and then in the West. G. Maruyama [40] and I. V. Girsanov [19] used stochastic integrals in order to describe the transformation of Wiener measure induced by an additional drift. First systematic expositions of stochastic integration and of stochastic differential On Kiyosi Itô’s work and its impact 117 equations appeared in E. B. Dynkin’s monograph [10] on Markov processes and, following earlier work of I. I. Gihman [16], [17] where some results of Itô had been found independently, in Gihman and Skorohod [18]. Kunita and Watanabe [34] clarified the geometry of spaces of martingales in terms of stochastic integrals. In the West, H. P. McKean published his book Stochastic Integrals [41] (dedicated to K. Itô) in 1969, and P.A. Meyer, C. Dellacherie, C. Doléans-Dade, J. Jacod and M.Yor started their systematic development of stochastic integration theory in the general framework of semimartingales; see, e.g. [8]. As a result, stochastic analysis emerged as one of the dominating themes of probability theory in the seventies. At the same time it began to interact increasingly with other mathematical fields. For example, J. Eells, K. D. Elworthy, P.Malliavin and others explored the idea of stochastic parallel transport presented by Kiyosi Itô at the ICM in Stockholm [28] and began to shape the new field of stochastic differential geometry; see, e.g., [12] and [13]. Connections to statistics, in particular to estimation and filtering problems for stochastic processes, were developed by R. S. Liptser and A. N. Shiryaev [35]. Infinite-dimensional extensions of stochastic analysis began to unfold in the eight- ies. Measure-valued diffusions and “superprocesses” arising as scaling limits of large systems of branching particles became an important area of research where the tech- niques of Itô calculus were crucial; see, e.g., [6], and [14]. Stochastic differential equations were studied in various infinite-dimensional settings, see, e.g., [1] and [5]. With his lectures Foundations of Stochastic Differential Equations in Infinite Dimen- sional Spaces [29], given at ETH Zurich and at Louisiana State University in 1983, Itô himself made significant contributions to this development. In fact, in his foreword to [24] Kiyosi Itô says that “it became my habit to observe even finite-dimensional facts from the infinite-dimensional viewpoint”. Paul Malliavin developed the stochas- tic analysis of an infinite-dimensional Ornstein–Uhlenbeck process and showed that this approach provides powerful new tools in order to obtain regularity results for the distributions of functionals of the solutions of stochastic differential equations [37]. His ideas led to what is now known as the Malliavin calculus, a highly sophisticated methodology with a growing range of applications which emerged in the eighties and nineties as one of the most important advances of stochastic analysis; see, e.g., [38] and [42]. While the impact of Itô’s ideas within mathematics took some time to become really felt, their importance was recognized early on in several areas outside of math- ematics. I will briefly mention some of them in anecdotical form before I describe one case study in more detail, namely the application of Itô’s calculus in finance. Already in the sixties engineers discovered that Itô’s calculus provides the right concepts and tools for analyzing the stability of dynamical systems perturbed by noise and to deal with problems of filtering and control. When I was an instructor at MIT in 1969/70, stochastic analysis did not appear in any course offered in the Department of Math- ematics. But I counted 4 courses in electrical engineering and 2 in aeronautics and astronautics in which stochastic differential equations played a role. The first sys- tematic exposition in Germany was the book Stochastische Differentialgleichungen 118 Hans Föllmer [2] by Ludwig Arnold, with the motion of satellites as a prime example. It was based on seminars and lectures at the Technical University Stuttgart which he was urged to give by his colleagues in engineering. In the seventies the relevance of Itô’s work was also recognized in physics and in particular in quantum field theory. When I came to ETH Zurich in 1977, Barry Simon gave a series of lectures for Swiss physicists on path integral techniques which included the construction of Itô’s integral for Brow- nian motion, an introduction to stochastic calculus, and applications to Schrödinger operators with magnetic fields; see chapter V in [45]. When Kiyosi Itô was awarded a honorary degree by ETH Zurich in 1987, this was in fact due to a joint initiative of mathematicians and physicists. In another important development, the methods of Itô’s calculus were crucial in analyzing scaling limits of models in population genet- ics in terms of measure-valued diffusions; see, e.g., [44] and the chapter on genetic models in [15], and [14]. I will now describe the application of Itô’s calculus in finance which began around 1970 and which has transformed the field in a spectacular manner, in parallel with the explosive growth of markets for financial derivatives. Consider the price fluctuation of some liquid financial asset, modeled as a stochastic process S = (St )0≤t≤T on some probability space ( , F ,P)with filtration (Ft )0≤t≤T . Usually S is assumed to be the solution of some stochastic differential equation (4), and then the volatility of the price fluctuation as measured by the quadratic variation process X is governed by the state-dependent diffusion coefficient σ(x) as described in equation (10). The best-known case is geometric Brownian motion, where the coefficients are of the form σ(x) = σx and b(x) = bx. This is known as the Black–Scholes model, and we will return to this special case below. In general, the choice of a specific model involves statistical and econometric considerations. But it also has theoretical aspects which are related to the idea of market efficiency. In its strong form, market efficiency requires that at each time t the available information and the market’s expectations are immediately “priced in”. Assuming a constant interest rate r, this means that the discounted price process X = (Xt )0≤t≤T defined by Xt = St exp(−rt) satisfies the condition E[Xt+s|Ft ]=Xt . In other words, the discounted price process is assumed to be a martingale under the given probability measure P , and in this case P is called a martingale measure with respect to the given price process. In this strong form market efficiency has a drastic consequence: There is no way to generate a systematic gain by using a dynamic trading strategy. This follows from Itô’s theory of the stochastic integral, applied to a general martingale instead of Brownian motion. Indeed, a trading strategy specifies the amount ξt of the underlying asset to be held at any time t. It is then natural to say that the resulting net gain at the final time T is given by Itô’s stochastic integral T V = ξ dX ≈ ξ (X − X ). T t t ti ti+1 ti 0 i On Kiyosi Itô’s work and its impact 119 Note in fact that the non-anticipating construction of the Itô integral matches exactly the economic condition that each investment decision is based on the available in- formation and is made before the future price increment is known. But if X is a martingale under the given probability measure P , as it is required by market effi- ciency in its strong form, then the stochastic integral inherits this property. Thus the expectation of the net gain under P is indeed given by E[VT ]=0. There is a much more flexible notion of market efficiency, also known as the “absence of arbitrage opportunities”. Here the existence of a trading strategy with positive expected net gain is no longer excluded. But it is assumed that there is no such profit opportunity without some downside risk, i.e., E[VT ] > 0 ⇒ P [VT < 0] = 0. As shown by Harrison and Kreps [20], and then in much greater generality by Delbaen and Schachermayer [7], this relaxed notion of market efficiency is equivalent to the condition that the measure P , although it may not be a martingale measure itself, does admit an equivalent martingale measure P ∗ ≈ P . Equivalent martingale measures provide the key to the problem of pricing and hedging financial derivatives. Such derivatives, also known as contingent claims, are financial contracts based on the underlying price process. The resulting discounted outcome can be described as a nonnegative random variable H on the probability space ( , FT ,P). The simplest example is a European call-option with maturity T , + where H = (XT − c) only depends on the value of the stock price at the final time T . A more exotic example is the look-back option given by the maximal stock price observed up to time T . For simple diffusion models such as the Black–Scholes model the equivalent martingale measure P ∗ is in fact unique, and in this case the financial market model is called complete. In such a complete situation any contingent claim H admits a unique arbitrage-free price, and this price is given by the expectation E∗[H] under the martingale measure P ∗. As shown by Jacod and Yor in the eighties, uniqueness of the equivalent martingale measure P ∗ is indeed equivalent to the fact that each contingent claim H admits a representation as a stochastic integral of the underlying price process: T ∗ H = E [H]+ ξt dXt . (14) 0 This result may in fact be viewed as an extension of a fundamental theorem of Itô on the representation of functionals of Brownian motion as stochastic integrals. For a simple diffusion model it is actually a direct consequence of Itô’s formula, as we will see below. In financial terms, the representation (14) means that the contingent claim H admits a perfect replication by means of a dynamic trading strategy, starting with 120 Hans Föllmer the initial capital E∗[H]. But this implies that the correct price is given by the initial capital, since otherwise there would be an obvious arbitrage opportunity. In the financial context, the crucial insight that arbitrage-free prices of derivatives should be computed as expectations under an equivalent martingale measure goes back to Black and Scholes [3]. They considered the problem of pricing a European call-option of geometric Brownian motion and realized that the key to the solution is provided by Itô’s formula. More generally, suppose that the price fluctuation is modeled by a stochastic differential equation (4) and that the contingent claim is of the form H = h(XT ) with some continuous function h. Note first that we can rewrite Itô’s formula (13) as ∂ df (X, t) =∇ f(X,t)dX+ L∗ + f(X,t)dt x ∂t ∗ in terms of the operator L = L − b∇x. Thus the contingent claim can be written as T H = f(x,0) + ∇xf(Xt ,t)dXt (15) 0 if the function f on Rd ×[0,T] is chosen to be a solution of the partial differential equation ∂ L∗ + f = ∂t 0 (16) with terminal condition f(·,T)= h. The representation (15) shows that the contin- gent claim admits a perfect replication, or a perfect hedge, by means of the strategy ∗ ξt =∇xf(Xt ,t). Therefore its arbitrage-free price is given by E [H]=f(x,0).In the same way, the arbitrage-free price at any time t is given by the value f(Xt ,t). Thus Itô’s formula provides an explicit method of computing the hedging strategy and the arbitrage-free price which involves the associated partial differential equation (16). This approach can be extended to arbitrarily exotic derivatives. Indeed, applying the preceding argument stepwise to products of the form H = hi(Xti ) and using an approximation of general derivatives by such finitely based functionals, one ob- tains the crucial representation (14) of a general contingent claim H as a stochastic integral of the underlying diffusion process. While this approach clarifies the pic- ture from a conceptual point of view, the explicit computation of the price and the hedging strategy usually becomes a major challenge when moving beyond the simple case of a call option. At this stage additional methods of numerical analysis and of stochastic analysis may be needed. In particular, the Malliavin calculus and the analysis of “cubature on Wiener space” developed by T. Lyons have started to play an important role in this context; see, e.g., Malliavin and Thalmaier [39] and Lyons and Victoir [36]. New conceptual problems arise as soon as the financial market model becomes incomplete, i.e., if the martingale measure P ∗ is no longer unique. This happens if, for example, the driving Brownian motion in (4) is replaced by a general Lévy process as in Itô’s original work, or if volatility becomes stochastic in the sense that On Kiyosi Itô’s work and its impact 121 the diffusion coefficient σ is replaced by a stochastic process. The issue of pricing and hedging financial derivatives in such an incomplete setting has led to new optimization problems and has opened new connections to convex analysis and to microeconomic theory. It has also become the source of new directions in martingale theory. In particular it has led to new variants of some fundamental decomposition theorems such as the Kunita–Watanabe decomposition and the Doob–Meyer decomposition, and it has motivated the systematic development of the theory of backward stochastic differential equations; see, e.g., [33] and [11]. In all these ramifications, however, Itô’s stochastic analysis continues to provide the crucial concepts and tools. In the beginning we recalled the statutes of the Gauss prize. We can now see more clearly why each and every one of their requirements is so well met by Kiyosi Itô’s contributions. In the first place, these contributions are outstanding and in fact of fundamental importance from a strictly mathematical point of view. Secondly, they have found significant applications outside of mathematics as illustrated by the preceding case study: There is no doubt that the field of quantitative finance has been thoroughly transformed by the basic insights provided by Itô’s calculus, both on a conceptual and on a computational level. Finally, this transformation of the field has paved the way to the innovative application of a wide range of mathematical methods, not only from stochastic analysis but also, following in their wake, methods from PDE’s, convex analysis, statistics, and numerical analysis. In their introduction to [24] quoted above, Stroock and Varadhan say that Kiyosi Itô “has molded the way in which we all think about stochastic processes”. When this was written, “we all” referred to a rather small group of specialists. Over the last three decades this group has increased dramatically, both within and beyond the boundaries of mathematics. And I am sure that there is overwhelming agreement with the anonymous weblog discussant that the Gauss prize has been awarded to “someone really wonderful”. References [1] Albeverio, S., and Röckner, M., Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Theory Relat. Fields 89 (1991), 347–386. 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Kyoto Univ. 21 (1) (1981), 133–151. [45] Simon, B., Functional Integration and Quantum Physics. Pure Appl. Math. 86, Academic Press, New York 1979. [46] Stroock, D. W., Markov Processes from K. Itô’s Perspective, Ann. of Math. Stud. 155, Princeton University Press, Princeton, NJ, 2003. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany E-mail: [email protected] Universality for mathematical and physical systems Percy Deift∗ Abstract. All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics. In this paper we describe some recent history of universality ideas in physics starting with Wigner’s model for the scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire called patience sorting. Mathematics Subject Classification (2000). Primary 60C05; Secondary 47N30. Keywords. Random matrices, universality, Riemann–Hilbert problems. 1. Introduction All physical systems in equilibrium obey the laws of thermodynamics. The first law asserts the conservation of energy. The second law has a variety of formulations, one of which is the following: Suppose that in a work cycle a heat engine extracts Q1 units of heat from a heat reservoir at temperature T1, performs W units of work, and then exhausts the remaining Q2 = Q1 − W units of heat to a heat sink at temperature W T η ≤ ηmax. (1) Nature is so set up that we just cannot do any better. On the other hand, it is a very old thought, going back at least to Democritus and the Greeks, that matter, all matter, is built out of tiny constituents – atoms – obeying their own laws of interaction. The juxtaposition of these two points of view, the ∗ The author would like to thank Sourav Chatterjee, Patrik Ferrari, Peter Sarnak and Toufic Suidan for useful comments and Irina Nenciu for her help and suggestions in preparing the manuscript. The work of the author was supported in part by DMS Grants No. 0296084 and No. 0500923, and also by a Friends of the Institute Visiting Membership at the Institute for Advanced Study in Princeton, Spring 2006. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2007 European Mathematical Society 126 Percy Deift macroscopic world of tangible objects and the microscopic world of atoms, presents a fundamental, difficult and long-standing challenge to scientists; namely, how does one derive the macroscopic laws of thermodynamics from the microscopic laws of atoms? The special, salient feature of this challenge is that the same laws of thermodynamics should emerge no matter what the details of the atomic interaction. In other words, on the macroscopic scale, physical systems should exhibit universality1. Indeed, it is the very emergence of universal behavior for macroscopic systems that makes possible the existence of physical laws. This kind of thinking, however, is not common in the world of mathematics. Mathematicians tend to think of their problems as sui generis, each with its own special, distinguishing features. Two problems are regarded as “the same” only if some isomorphism, explicit or otherwise, can be constructed between them. In recent years, however, universality in the above sense of macroscopic physics has started to emerge in a wide variety of mathematical problems, and the goal of this paper is to illustrate some of these developments. As we will see, there are problems from diverse areas, often with no discernible, mechanistic connections, all of which behave, on the appropriate scale, in precisely the same way. The list of such problems is varied, long and growing, and points to the emergence of what one might call “macroscopic mathematics.” A precedent for the kind of results that we are going to describe is given by the celebrated central limit theorem of probability theory, where one considers indepen- {x } dent, identically distributed variables n n≥1. The central limit √ theorem tells us that if we center and scale the variables, xn → yn ≡ xn − E(xn) / V(xn), then n t u2 k= yk − du lim Prob √1 ≤ t = e 2 √ . (2) n→∞ n −∞ 2π We see here explicitly that the Gaussian distribution on the right-hand side of (2) is universal, independent of the distribution for the xn’s. The proof of the central limit (x =+) = (x =−) = 1 theorem for independent coin flips, Prob n 1 Prob n 1 2 , goes back to de Moivre and Laplace in the 18th century. Of course (2) is only one of many similar universality-type results now known in probability theory. The outline of the paper is as follows: In Section 2 we will introduce and discuss some models from random matrix theory (RMT). Various distributions associated with these models will play the same role in the problems that we discuss later on in the paper as the Gaussian does in (2). As noted above, thermodynamics reflects universality for all macroscopic systems, but there are also many universality sub- classes which describe the behavior of physical systems in restricted situations. For example, many fluids, such as water and vinegar, obey the Navier–Stokes equation, but a variety of heavy oils obey the lubrication equations. In the same way we will see 1In physics, the term “universality” is usually used in the more limited context of scaling laws for critical phenomena. In this paper we use the term “universality” more broadly in the spirit of the preceding discussion. We trust this will cause no confusion. Universality for mathematical and physical systems 127 that certain mathematical problems are described by so-called Unitary Ensembles of random matrices, and others by so-called Orthogonal or Symplectic Ensembles. In Section 3, we present a variety of problems from different areas of mathematics, and in Section 4 we show how these problems are described by random matrix models from Section 2. In the final Section 5 we discuss briefly some of the mathematical methods that are used to prove the results in Section 4. Here combinatorial identities, Riemann–Hilbert problems (RHP’s) and the nonlinear steepest descent method of [DeiZho], as well as the classical steepest descent method, play a key role. We end the section with some speculations, suggesting how to place the results of Sections 3 and 4 in a broader mathematical framework. 2. Random matrix models There are many ensembles of random matrices that are of interest, and we refer the reader to the classic text of Mehta [Meh] for more information (see also [Dei1]). In this paper we will consider almost exclusively (see, however, (54) et seq. below) only three kinds of ensembles: (a) Orthogonal Ensembles (OE’s) consisting of N × N real symmetric matrices M, M = M¯ = MT . (b) Unitary Ensembles (UE’s) consisting of N × N Hermitian matrices M, M = M∗. (c) Symplectic Ensembles (SE’s) consisting of 2N × 2N Hermitian, self-dual ∗ T T matrices M = M = JM J , where J is the standard 2N × 2N block J = (τ,τ,...,τ) τ = 01 diagonal symplectic matrix, diag , −10 . For reasons that will soon become clear, OE’s, UE’s and SE’s are labeled by a pa- rameter β, where β = 1, 2 or 4, respectively. In all three cases the ensembles are equipped with probability distributions of the form 1 − tr VN,β(M) PN,β(M) dβ M = e dβ M (3) ZN,β where VN,β is a real-valued function on R such that VN,β(x) →+∞sufficiently rapidly as |x|→∞, ZN,β is a normalization coefficient, and dβ M denotes Lebesgue measure on the algebraically independent entries of M. For example, in the orthogonal d M = dM M = (M ) case, β=1 1≤j≤k≤N jk, where jk (see, e.g. [Meh]). The notation “orthogonal”, “unitary”, and “symplectic” refers to the fact that the above ensem- bles with associated distributions (3) are invariant under conjugation M → SMS−1, where S is orthogonal, unitary, or unitary-symplectic (i.e., S ∈ USp(2N) ={S : ∗ T SS = I, SJS = J }) respectively. When VN,β(x) = x2, one has the so-called Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE), and the Gaussian Symplectic Ensemble (GSE), for β = 1, 2 or 4, respectively. 128 Percy Deift The distributions (3) in turn give rise to distributions on the eigenvalues λ1 ≤ λ2 ≤··· of M, N ˆ N 1 −ηβ i= VN,β(λi ) β PN,β(λ) d λ = e 1 |λi − λj | dλ1 ···dλN (4) Zˆ N,β 1≤i For energies E at the edge of the spectrum, the above procedure must be modified slightly (see below). This procedure can be viewed as follows: A scientist wishes to investigate some statistical phenomenon. What s’he has at hand is a microscope and a handbook of matrix ensembles. The data {ak} are embedded on a slide which can be inserted into the microscope. The only freedom that the scientist has is to center the slide, ak → ak −A, and then adjust the focus ak −A →˜ak = γa(ak −A) so that on average one data point a˜k appears per unit length on the slide. At that point the scientist takes out his’r handbook, and then tries to match the statistics of the a˜k’s with those of the eigenvalues of some ensemble. If the fit is good, the scientist then says that the system is well-modeled by RMT. It is a remarkable fact, going back to the work of Gaudin and Mehta, and later Dyson, in the 1960s, that the key statistics for OE’s, UE’s, and SE’s can be computed in closed form. This is true not only for finite N, but also for various scaling limits as N →∞. For GOE, GUE, and GSE we refer the reader to [Meh]. Here the Hermite 2 polynomials, which are orthogonal with respect to the weight e−x dx on R, play Universality for mathematical and physical systems 129 a critical role, and the scaling limits as N →∞follow from the known, classical asymptotics of the Hermite polynomials. For UE’s with general potentials VN,β=2, the techniques described in [Meh] for GUE go through for finite N, the role of the Hermite polynomials now being played by the polynomials orthogonal with respect −VN,β= (x) to the weight e 2 dx on R (see, e.g. [Dei1]). For general VN,β=2, however, the asymptotic behavior of these polynomials as N →∞does not follow from classical estimates. In order to overcome this obstacle, the authors in [DKMVZ1] and [DKMVZ2] (see also [Dei1] for a pedagogical presentation) used the Riemann– Hilbert steepest-descent method introduced by Deift and Zhou [DeiZho], and further developed with Venakides [DVZ], to compute the asymptotics as N →∞of the orthogonal polynomials for a very general class of analytic weights. In view of the preceding comments, the scaling limits of the key statistics for UE’s then follow for 4 2 such weights (see also [BleIts] for the special case VN,β=2(x) = N(x − tx )). For another approach to UE universality, see [PasSch]. For OE’s and SE’s with classical weights, such as Laguerre, Jacobi, etc., for which the asymptotics of the associated orthogonal polynomials are known, the GOE and GSE methods in [Meh] apply (see the introductions to [DeiGio1] and [DeiGio2] for a historical discussion). For general VN,β, β = 1 or 4, new techniques are needed, and these were introduced, for finite N, by Tracy and Widom in [TraWid2] and [Wid]. In [DeiGio1] and [DeiGio2], the authors use the results in [TraWid2] and [Wid], together with the asymptotic estimates in [DKMVZ2], to compute the large N limits of the key statistics for OE’s and SE’s 2m with general polynomial weights VN,β(x) = κ2mx +···, κ2m > 0. It turns out that not only can the statistics for OE’s, UE’s and SE’s be computed explicitly, but in the large N limit the behavior of these systems is universal in the sense described above, as conjectured earlier by Dyson, Mehta, Wigner, and many others. It works like this: Consider N × N matrices M in a UE with potential VN,2. Let KN denote the finite rank operator with kernel N −1 KN (x, y) = ϕj (x)ϕj (y), x, y ∈ R (7) j=0 where − 1 V (x) ϕj (x) = pj (x)e 2 N,2 ,j≥ 0 (8) and j pj (x) = γj x +··· ,j≥ 0,γj > 0 (9) −V (x) are the orthonormal polynomials with respect to the weight e N,2 dx, −V (x) pj (x)pk(x)e N,2 dx = δjk,j,k≥ 0. R Then the m-point correlation functions N! R (λ ,...,λ ) ≡ ··· Pˆ (λ ,...,λ )dλ ···dλ m 1 m (N − m)! N,2 1 N m+1 N 130 Percy Deift can be expressed in terms of KN as follows: Rm(λ ,...,λm) = KN (λi,λj ) . 1 det 1≤i,j≤m (10) A simple computation for the 1-point and 2-point functions, R1(λ) and R2(λ1,λ2), shows that E #{λi ∈ B} = R1(λ) dλ (11) B for any Borel set B ⊂ R, and E #{ordered pairs (i, j), i = j : (λi,λj ) ∈ } = R2(λ1,λ2)dλ1dλ2 (12) for any Borel set ⊂ R2. It follows in particular from (11) that, for an energy E, R1(E) = KN (E, E) is the density of the expected number of eigenvalues in a neighborhood of E, and hence, by the standard procedure, one should take the scaling factor γλ in (5) to be KN (E, E). For energies E in the bulk of the spectrum, one finds for a broad class of potentials VN,2 (see [DKMVZ1] and [DKMVZ2]) that, in the scaling limit dictated by KN (E, E), KN (λ, λ ) takes on a universal form 1 x y lim KN E + ,E+ = K∞(x − y) (13) N→∞ KN (E, E) KN (E, E) KN (E, E) where x,y ∈ R and K∞ is the so-called sine-kernel, (πu) K (u) = sin . ∞ πu (14) Inserting this information into (10) we see that the scaling limit for Rm is universal for each m ≥ 2, and in particular for m = 2, we have for x,y ∈ R 1 x y lim R E + ,E+ N→∞ 2 2 K (E, E) K (E, E) KN (E, E) N N K∞(0)K∞(x − y) = det (15) K∞(x − y) K∞(0) π(x − y) 2 = − sin . 1 π(x − y) For a Borel set B ⊂ R, let nB = #{λj : λj ∈ B} and let 2 VB = E nB − E(nB ) (16) denote the number variance in B. A simple computation again shows that 2 VB = R1(x) dx + R2(x,y)dxdy− R1(x) dx . B B×B B Universality for mathematical and physical systems 131 For an energy E in the bulk of the spectrum as above, set s s BN (s) = E − ,E+ ,s>0. 2KN (E, E) 2KN (E, E) For such B, VB is the number variance for an interval about E of scaled size s. Recalling that KN (E, E) = R1(E), and using (15), we find as N →∞ 1 2πs 1 − cos u 2s ∞ sin u 2 VB (s) = du + du. lim N 2 (17) N→∞ π 0 u π πs u For large s, the right-hand side has the form (see [Meh]) 1 1 log(2πs)+ γ + 1 + O (18) π 2 s where γ is Euler’s constant. For θ>0, the so-called gap probability GN,2(θ) = Prob M : M has no eigenvalues in (E − θ,E + θ) (19) is given by (see [Meh], and also [Dei1]) GN,2(θ) = det 1 − KNL2(E−θ,E+θ) (20) 2 where KN L2(E−θ,E+θ) denotes the operator with kernel (7) acting on L (E − θ, E + θ). In the bulk scaling limit, we find x lim GN,2 = det 1 − K∞L2(E−θ,E+θ) ,x∈ R. (21) N→∞ KN (E, E) ˜ In terms of the scaled eigenvalues λj = KN (E, E) · (λj − E), this means that for x>0 ˜ lim Prob M : λj ∈/ (−x,x), 1 ≤ j ≤ N = det 1 − K∞L2(−x,x) . (22) N→∞ Now consider a point E, say E = 0, where KN (E, E) = KN (0, 0) →∞as N →∞. This is true, in particular, if 2m VN,2(x) = κmx +··· ,κm > 0,m≥ 1, (23) 2 1− 1 and so for VN,2(x) = x (GUE). For such VN,2’s, we have KN (0, 0) ∼ N 2m (see [DKMVZ1]). Let tN > 0 be such that tN tN →∞, → 0. (24) KN (0, 0) 132 Percy Deift Then tN ˆ tN KN (0,0) N ≡ E # |λj |≤ = KN (λ,λ)dλ∼ 2tN →∞. (25) KN (0, 0) − tN KN (0,0) For a Then we have by (12) and (15), as N →∞, 1 E { (i, j), i = j : (λ ,λ ) ∈ } ˆ # ordered pairs i j N N = 1 R (λ ,λ )dλ dλ Nˆ 2 1 2 1 2 N 1 1 s t = R , ds dt ˆ 2 2 K ( , ) K ( , ) N {(s,t): a Another quantity of interest is the spacing distribution of the eigenvalues λ1 ≤ λ2 ≤ ··· ≤ λN of a random N × N matrix as N →∞. More precisely, for s>0, we want to compute #{1 ≤ j ≤ N − 1 : λj+ − λj ≤ s} E 1 N as N becomes large. If we again restrict our attention to eigenvalues in a neighborhood of a bulk energy E = 0, say, then the eigenvalue spacing distribution exhibits universal behavior for UE’s as N →∞. We have in particular the following result of Gaudin Universality for mathematical and physical systems 133 ˆ (see [Meh], and also [Dei1]): With tN , N and x˜j = KN (0, 0)xj as above, ˜ ˜ ˜ #{1 ≤ j ≤ N − 1 : λj+ − λj ≤ s, |λj |≤tN } lim E 1 N→∞ Nˆ ˜ = lim Prob(at least one eigenvalue λj in (0,s]|eigenvalue at 0) (28) N→∞ s = p(u) du 0 where d2 p(u) = det 1 − K∞ 2 . (29) du2 L (0,u) At the upper spectral edge E = λmax, one again finds universal behavior for UE’s with potentials VN,2, in particular, of the form (23) above. For such VN,2’s there exist (2) (2) constants zN , sN such that for t ∈ R (see [DeiGio2] and the notes therein) (2) λmax − zN lim Prob M : ≤ t = det 1 − AL2(t,∞) . (30) N→∞ (2) sN Here A is the so-called Airy operator with kernel (x) (y) − (x) (y) A(x, y) = Ai Ai Ai Ai , x − y (31) (x) V (x) = x2 where Ai√ is the classical Airy function. For GUE, where N,2 , one has (2) (2) − 1 − 1 zN = 2N and sN = 2 2 N 6 (see Forrester [For1] and the seminal work of Tracy and Widom [TraWid1]). It turns out that det 1 − K∞ L2(−x,x) in (21) and det 1 − AL2(t,∞) can be expressed in terms of solutions of the PainlevéV and Painlevé II equations respectively. The first is a celebrated result of Jimbo, Miwa, Môri, and Sato [JMMS], and the second is an equally celebrated result of Tracy and Widom [TraWid1]. In particular for edge scaling we find (2) λmax − zN lim Prob M : ≤ t = Fβ= (t) (32) N→∞ (2) 2 sN where ∞ 2 − t (s−t)u (s) ds Fβ=2(t) = det 1 − AL2(t,∞) = e (33) and u(s) is the (unique, global) Hastings–McLeod solution of the Painlevé II equation u (s) = 2u3(s) + su(s) (34) such that u(s) ∼ Ai(s) as s →+∞. (35) 134 Percy Deift F2(t) = Fβ=2(t) is called the Tracy–Widom distribution for β = 2. Finally we note that for OE’s and SE’s there are analogs for all the above results (10)–(35), and again one finds universality in the scaling limits as N →∞for potentials VN,β, β = 1, 4, of the form (23) above (see [DeiGio1] and [DeiGio2] and the historical notes therein). We note, in particular, the following results: for VN,β as (β) (β) above, β = 1 or 4, there exist constants zN ,sN such that (β) λmax(M) − zN lim Prob M : ≤ t = Fβ (t) (36) N→∞ (β) sN where 1 1 ∞ 2 − t u(s) ds F1(t) = F2(t) e 2 (37) and 1 ∞ u(s) ds − 1 ∞ u(s) ds 1 e 2 t + e 2 t F (t) = F (t) 2 · (38) 4 2 2 with F2(t) and u(s) as above. F1(t) and F4(t) are called the Tracy–Widom distribu- tions for β = 1 and 4 respectively. 3. The problems In this section we consider seven problems. The first is from physics and is included for historical reasons that will become clear in Section 4 below; the remaining six problems are from mathematics/mathematical physics. Problem 1. Consider the scattering of neutrons off a heavy nucleus, say uranium U238. The scattering cross-section is plotted as a function of the energy E of the incoming neutrons, and one obtains a jagged graph (see [Por] and [Meh]) with many hundreds of sharp peaks E1 (ii) in all cases, the Ej ’s exhibited repulsion, or, more precisely, the probability that two Ej ’s would be close together was small. Question 1. What theory did Wigner propose for the Ej ’s? Problem 2. Here we consider the work of H. Montgomery [Mon] in the early 1970s on the zeros of the Riemann zeta function ζ(s). Assuming the Riemann hypothesis, Montgomery rescaled the imaginary parts γ1 ≤ γ2 ≤ ··· of the (nontrivial) zeros { 1 + iγ } ζ(s) 2 j of , γj log γj γj →˜γj = (39) 2π to have mean spacing 1 as T →∞, i.e. #{j ≥ 1 :˜γj ≤ T } lim = 1. T →∞ T For any a Question 3. Equip SN with the uniform distribution. If each card is of unit size, how big a table does one typically need to play patience sorting with N cards? Or, more precisely, how does pn,N = Prob π : qN (π) ≤ n (41) behave as N →∞, n ≤ N? Problem 4. The city of Cuernavaca in Mexico (population about 500,000) has an extensive bus system, but there is no municipal transit authority to control the city transport. In particular there is no timetable, which gives rise to Poisson-like phe- nomena, with bunching and long waits between buses. Typically, the buses are owned by drivers as individual entrepreneurs, and all too often a bus arrives at a stop just as another bus is loading up. The driver then has to move on to the next stop to find his fares. In order to remedy the situation the drivers in Cuernavaca came up with a novel solution: they introduced “recorders” at specific locations along the bus routes in the city. The recorders kept track of when buses passed their locations, and then sold this information to the next driver, who could then speed up or slow down in order to optimize the distance to the preceding bus. The upshot of this ingenious scheme is that the drivers do not lose out on fares and the citizens of Cuernavaca now have a reliable and regular bus service. In the late 1990s two Czech physicists with interest in transportation problems, M. Krbálek and P.Šeba, heard about the buses in Cuernavaca and went down to Mexico to investigate. For about a month they studied the statistics of bus arrivals on Line 4 close to the city center. In particular, they studied the bus spacing distribution, and also the bus number variance measuring the fluctuations of the total number of buses arriving at a fixed location during a time interval T . Their findings are reported in [KrbSeb]. Question 4. What did Krbálek and Šeba learn about the statistics of the bus system in Cuernavaca? Problem 5. In his investigation of wetting and melting phenomena in [Fis], Fisher introduced various “vicious” walker models. Here we will consider the so-called random turns vicious walker model. In this model, the walks take place on the integer lattice Z and initially the walkers are located at 0, 1, 2,.... The rules for a walk are as follows: (a) at each tick of the clock, precisely one walker makes a step to the left; (b) no two walkers can occupy the same site (hence “vicious walkers”). For example, consider the following walk from time t = 0 to time t = 4: At t = 0, clearly only the walker at 0 can move. At time t = 1, either the walker at −1orat+1 can move, and so on. Let dN be the distance traveled by the walker starting from 0. In the above example, d4 = 2. For any time N, there are clearly only a finite number of possible walks of duration t = N. Suppose that all such walks are equally likely. Question 5. How does dN behave statistically as N →∞? Universality for mathematical and physical systems 137 · ·×·××·×× t = 4 · ·×·×·××× t = 3 · · · × × · ××× t = 2 · · · × · ×××× t = 1 · · · · ××××× t = 0 −4 −3 −2 −10 1 2 3 4 Figure 1. Random turns walk. Problem 6. Consider tilings {T } of the tilted square Tn ={(x, y) :|x|+|y|≤n+1} in R2 by horizontal and vertical dominos of length 2 and width 1. For example, for n = 3 we have the tiling T of Figure 2. For each tiling the dominos must lie strictly Figure 2. Aztec diamond for n = 3. within Tn. The tilings T are called Aztec diamonds because the boundary of T in {(x, y) : y>0}, say, has the shape of a Mexican pyramid. It is a nontrivial theorem n(n+1) (see [EKLP]) that for any n, the number of domino tilings of Tn is 2 2 . Assume that all tilings are equally likely. Question 6. What does a typical tiling look like as n →∞? Finally we have Problem 7. How long does it take to board an airplane? We consider the random boarding strategy in [BBSSS] under the following simplifying assumptions: (a) there is only 1 seat per row; (b) the passengers are very thin compared to the distance between seats; (c) the passengers move very quickly between seats. The main delay in boarding is the time – one unit – that it takes for the passengers to organize their luggage and seat themselves once they arrive at their assigned seats. 138 Percy Deift For the full problem with more than one seat per row, passengers who are not “very thin”, etc., see [BBSSS], and also the discussion of the boarding problem in Section 4 below. The passengers enter the airplane through a door in front and the seats are numbered 1, 2,...,N, with seat 1 closest to the door. How does boarding proceed? Consider, for example, the case N = 6. There are 6 passengers, each with a seating card 1, 2,...,6. At the call to board, the passengers line up randomly at the gate. Suppose for definitiveness that the order in the line is given by π : 341562 (42) with 3 nearest the gate. Now 3 can proceed to his’r seat, but 4 is blocked and must wait behind 3 while s’he puts his’r bag up into the overhead bin. However, at the same time, 1 can proceed to his’r seat, but 5, 6, and 2 are blocked. At the end of one unit of time, 3 and 1 sit down, and 4 and 2 can proceed to their seats, but 5 and 6 are blocked behind 4. After one more unit of time 4 and 2 sit down, and 5 can proceed to his’r seat, but 6 is blocked. At the end of one more unit of time 5 sits down, and finally 6 can move to his’r seat. Thus for π as above, it takes 4 units of time to board. Let bN = bN (π) denote the boarding time for any π ∈ SN , and assume that the π’s are uniformly distributed. Question 7. How does bN (π) behave statistically as N →∞? 4. Solutions and explanations As indicated in the Introduction, the remarkable fact of the matter is that all seven problems in Section 3 are modeled by RMT. Problem 1 (Neutron scattering). At some point in the mid-1950s, in a striking devel- opment, Wigner suggested that the statistics of the neutron scattering resonances was governed by GOE2 (and hence, by universality [DeiGio1], by all OE’s). And indeed, if one scales real scattering data for a variety of nuclei according to the standard pro- cedure and then evaluates, in particular, the nearest neighbor distribution, one finds remarkable agreement with the OE analog of the spacing distribution (28), (29). It is interesting, and informative, to trace the development of ideas that led Wigner to his suggestion (see [Wig1], [Wig2], [Wig3]; all three papers are reproduced in [Por]). In these papers, Wigner is guided by the fact that any model for the reso- nances would have to satisfy the constraints of universality and repulsion, (i) and (ii) respectively, in the description of Problem 1. In [Wig2] he recalls a paper that he had written with von Neumann in 1929 in which they showed, in particular, that in the n(n+1) n × n 2 -dimensional space of real symmetric matrices, the matrices with double 2Here we must restrict the data to scattering for situations where the nuclear forces are time-reversal invariant. If not, the statistics of the scattering resonances should be governed by GUE. Universality for mathematical and physical systems 139 eigenvalues form a set of codimension 2. For example, if a 2 × 2 real symmetric matrix has double eigenvalues, then it must be a multiple of the identity and hence it lies in a set of dimension 1 in R3. It follows that if one equips the space of real, symmetric matrices with a probability measure with a smooth density, the probability of a matrix M having equal eigenvalues would be zero and the eigenvalues λ1,...,λn of M would comprise a random set with repulsion built in. So Wigner had a model, or more precisely, a class of models, which satisfied constraint (ii). But why choose GOE? This is where the universality constraint (i) comes into play. We quote from [Wig3]3: “Let me say only one more word. It is very likely that the curve in Figure 1 is a universal function. In other words, it doesn’t depend on the details of the model with which you are working. There is one particular model in which the probability of the energy levels can be written down exactly. I mentioned this distribution already in Gatlinburg. It is called the Wishart distribution. Consider a set….” So in this way Wigner introduced GOE into theoretical physics: It provided a model with repul- sion (and time-reversal) built in. Furthermore, the energy level distribution could be computed explicitly. By universality, it should do the trick! As remarkable as these developments were, even the most prophetic observer could not have predicted that, a few years down the line, these developments would make themselves felt within pure mathematics. Problem 2 (Riemann zeta function). Soon after completing his work on the scaling limit (40) of the two-point correlation function for the zeros of zeta, Montgomery was visiting the Institute for Advanced Study in Princeton and it was suggested that he show his result to Dyson. What happened is a celebrated, and oft repeated, story in the lore of the Institute: before Montgomery could describe his hard won result to Dyson, Dyson took out a pen, wrote down a formula, and asked Montgomery “And did you get this?” b sin(πr) 2 R(a,b) = 1 − dr (43) a πr Montgomery was stunned: this was exactly the formula he had obtained. Dyson explained: “If the zeros of the zeta function behaved like the eigenvalues of a random GUE matrix, then (43) would be exactly the formula for the two-point correlation function!” (See (27) above.) More precisely, what Montgomery actually proved was that 1 sin πr 2 lim f(γ˜i −˜γj ) = f(r) 1 − dr (44) N→∞ N R πr 1≤i =j≤N for any rapidly decaying function f whose Fourier transform f(ξ)ˆ is supported in the interval |ξ| < 2. Of course, if one could prove (44) for all smooth, rapidly decaying functions, one would recover the full result (43). Nevertheless, in an impressive series 3In the quotation that follows, “Figure 1” portrays a level spacing distribution, the “Wishart distribution” is the statisticians’ name for GOE, and “Gatlinburg” is [Wig1]. 140 Percy Deift of numerical computations starting in the 1980s, Odlyzko verified (43) to extraordi- nary accuracy (see [Odl1], [Odl2], and the references therein). In his computations, Odlyzko also considered GUE behavior for other statistics for the γ˜j ’s, such as the nearest neighbor spacing, verifying in particular the relationship 1 s lim #{1 ≤ j ≤ N − 1 :˜γj+1 −˜γj ≤ s}= p(u) du, s > 0, (45) N→∞ N 0 (cf. (28), (29)) to high accuracy. The relationship between the zeros of the zeta function and random matrix theory first discovered by Montgomery has been taken up with great virtuosity by many researchers in analytic number theory, with Rudnick and Sarnak [RudSar], and then Katz and Sarnak [KatSar], leading the way. GUE behavior for the zeros of quite general automorphic L-functions over Q, as well as for a wide class of zeta and L-functions over finite fields, has now been established (modulo technicalities as in (44) above in the number field case). Another major development has been the discovery of a relationship between random polynomials whose roots are given by the eigenvalues of a matrix from some random ensemble, and the moments of the z = 1 L-functions on the critical line Re 2 (see [KeaSna1], [KeaSna2]). The discovery of Montgomery/Odlyzko counts as one of the major developments in analytic number theory in many, many years. Problem 3 (Patience sorting). In 1999 Baik, Deift and Johansson [BDJ1] proved the q (π) following result for N , the number of piles√ obtained in patience sorting starting qN −2 N from a shuffle π of N cards. Let χN = / . Then N1 6 lim Prob χN ≤ t = F (t) (46) N→∞ 2 where F2 is the Tracy–Widom distribution (32), (33) for β = 2. Thus the number of piles, suitably centered and scaled, behaves statistically like the largest eigenvalue of a GUE matrix. In addition, the authors proved convergence of moments. For any m = 1, 2,..., m m lim E(χN ) = E(χ ) (47) N→∞ where χ is any random variable with distribution F2. In particular, for m = 1, 2 one obtains √ E(q ) − N N 2 = tdF (t) lim 1/6 2 (48) N→∞ N R and V(q ) 2 N = t2 dF (t) − tdF (t) . lim 1/3 2 2 (49) N→∞ N R R Numerical evaluation shows that the constants on the right-hand side of (48) and (49) are given by −1.7711 and 0.8132, respectively. Thus, as N →∞, √ 1/6 E(qN ) ∼ 2 N − 1.7711 · N Universality for mathematical and physical systems 141 so that for a deck of N = 52 cards, one needs a table of size about 12 units on average to play the game. Patience sorting is closely related to the problem of longest increasing subse- quences for permutations π ∈ SN . Recall that we say that π(i1),...,π(ik) is an increasing subsequence in π of length k if i1 Consequently, the number of boxes in the first row of Plancherel-random Young diagrams behaves statistically, as N →∞, like the largest eigenvalue of a GUE matrix. In [BDJ1] the authors conjectured that the number of boxes in the first k rows of a Young diagram should behave statistically as N →∞like the top k eigenvalues λN ≥ λN−1 ≥ ··· ≥ λN−k+1 of a GUE matrix. This conjecture was proved for the 2nd row in [BDJ2]. For general k, the conjecture was proved, with convergence in joint distribution, in three separate papers in rapid succession ([Oko], [BOO], [Joh1]), using very different methods. The proof in [Oko] relies on an interplay between maps on surfaces and ramified coverings of the sphere; the proof in [BOO] is based on the analysis of specific characters on S(∞), the infinite symmetric group defined as the inductive limit of the finite symmetric groups SN under the embeddings SN → SN+1; and the proof in [Joh1] utilizes certain discrete orthogonal polynomial ensembles arising in combinatorial probability. One can consider the statistics of lN (π) for π restricted to certain distinguished subsets of SN (see [BaiRai1]). Amongst the many results in [BaiRai1] relating com- (inv) binatorics and random matrix theory, we mention the following. Let SN ={π ∈ SN : π 2 = id} be the set of involutions in SN . Then, under the Robinson–Schensted (inv) correspondence, uniform distribution on SN pushes forward to a new measure on 142 Percy Deift Young diagrams, different from the Plancherel measure. Denote this measure by Prob(inv), and in place of (51) we have (inv) (inv) Prob π ∈ SN : lN (π) ≤ n = Prob μ N : μ1 ≤ n . In [BaiRai1] the authors show that √ lN − 2 N lim Prob π ∈ S(inv) : ≤ x N→∞ N N 1/6 √ μ − 2 N (52) = lim Prob(inv) μ N : 1 ≤ x N→∞ N 1/6 = Fβ=1(x) and for the second row of μ √ μ − N (inv) 2 2 lim Prob μ N : ≤ x = Fβ= (x). (53) N→∞ N 1/6 4 Here Fβ=1 and Fβ=4 are the Tracy–Widom distributions for the largest eigenvalue of the GOE and GSE ensemble, respectively (see (36), (37), (38)). Thus all three of the basic ensembles β = 1, 2 and 4 show up in the analysis of the (general) increasing subsequence problem. A problem which is closely related to the longest increasing subsequence problem is the random word problem. In [TraWid4] the authors consider words {ω} of length N in an alphabet of k letters, i.e. maps ω :{1, 2,...,N}→{1, 2,...,k}. One says that ω(i1),...,ω(ij ) is a weakly increasing subsequence in ω of length j if wk i1 wk N lN (ω) − k lim Prob ω : ≤ s N→∞ 2N k (54) k 2 − i= xi 2 = γk e 1 (xi − xj ) dx1 ···dxk−1 L s 1≤i It is easy to see that the right-hand side of (54) is just the distribution function for the largest eigenvalue of a k × k GUE matrix conditioned to have trace zero. Consider the representation of the number π, say, in any basis b, q π = 0.a1a2a3 ...× b ,q∈ Z. (57) It has long been believed that in some natural asymptotic sense the digits a1, a2, a3,… are independent and identically distributed, with uniform distribution on {0, 1,...,b− 1}. In an attempt to formalize this notion, E. Borel (1909) introduced the idea of normality (see [Wag]): A real number x is normal if for any base b,any m ≥ 1, and any m-string s, #{occurrences of s in the first n base-b digits of x} −m lim = b . (58) n→∞ n While it is known that non-normal numbers form a set of Lebesgue measure zero, and all numerical evidence confirms (58) to high order, no explicit examples of normal numbers are known. Relation (54) suggests a new way to test for asymptotic randomness, as follows. Consider the first LN base-b digits a1a2 ...aLN of a given number x, where L and N are “large”. Partition these digits into L words ωj = a(j−1)N+1 ...ajN,1≤ j ≤ L, wk each of length N. For each ωj compute lN (ωj ). Then if the digits {aj } of x are asymptotically random, we could expect that as L, N →∞, the empirical distribution wk N 1 lN (ωj ) − b # 1 ≤ j ≤ L : ≤ s L 2N b is close to the conditional GUE distribution on the right-hand side of (54). Preliminary calculations in [DeiWit] for x = π and b = 2 show that for L, N “large” the empirical distribution is indeed close to the right-hand side of (54) with high accuracy. The work is in progress. Problem 4 (Bus problem in Cuernavaca). Krbálek and Šeba found that both the bus spacing distribution and the number variance are well modeled by GUE, (28), (29) and (17) respectively (see Figures 2 and 3 in [KrbSeb]). In order to provide a plausible explanation of the observations in [KrbSeb], the authors in [BBDS] introduced a microscopic model for the bus line that leads simply and directly to GUE. The main features of the bus system in Cuernavaca are (a) the stop-start nature of the motion of the buses; (b) the “repulsion” of the buses due to the presence of recorders. To capture these features, the authors in [BBDS] introduced a model for the buses consisting of n(= # of buses) independent, rate 1 Poisson processes moving from the bus depot at time t = 0 to the final terminus at time T , and conditioned not to intersect 144 Percy Deift for 0 ≤ t ≤ T . The authors then showed that at any observation point x along the route of length N>n, the probability distribution for the (rescaled) arrival times of 2tj the buses, yj = T − 1 ∈[−1, 1],1≤ j ≤ n,isgivenby n 2 const. wJ (yj ) (yi − yj ) dy1 ···dyn (59) j=1 1≤i Problem 6 (Aztec diamond). After scaling by n+1, Jockush et al., [JPS], considered 2 × 1 T ={(u, v) : the tiling problem with dominos of size n+1 n+1 in the tilted square 0 |u|+|v|≤1}.Asn →∞, they found that the inscribed circle C0 ={(u, v) : u2 + v2 = 1 } 2 , which they called the arctic circle, plays a remarkable role. In the four regions of T 0 outside C0, which they call the polar regions and label N, E, S, W clockwise from the top, the typical tiling is frozen, with all the dominoes in N and S horizontal, and all the dominos in E and W vertical. In the region inside C0, which they call the temperate zone, the tiling is random. (See, for example, http:// www.math.wisc.edu/~propp/tiling, where a tiling with n = 50 is displayed.) But more is true. In [Joh1], [Joh2], Johansson considered fluctuations of the boundary of the temperate zone about the circle C0. More precisely, for −1 <α<1, α = 0, let √ √ √ √ (x+,y+) = α+ 1−α2 , α− 1−α2 ,(x−,y−) = α− 1−α2 , α+ 1−α2 α α 2 2 α α 2 2 u + v = α C ={u2 + v2 = 1 } denote the two points of intersection of the line with 0 2 . Then for fixed α, Johansson showed that the fluctuations of the boundary of the tem- + + − − perate zone along the line u + v = α about the points (xα ,yα ) and (xα ,yα ) were described by the Tracy–Widom distribution F2 (see [Joh2], equation (2.72), for a precise statement). Johansson proceeds by expressing the fluctuations in terms of the Krawtchouk ensemble (cf. Problem 4), which he then evaluates asymptotically as n →∞. Such an analysis is possible because the associated Krawtchouk polynomials have an integral representation which can be evaluated asymptotically using the clas- sical method of steepest descent. In [CLP], the authors considered tilings of hexagons of size n by unit rhombi and proved an arctic circle theorem for the tilings as n →∞ as in the case of the Aztec diamond. In [Joh1], [Joh2], Johansson again expressed the fluctuations of the arctic circle for the hexagons in terms of a random particle ensem- ble, but now using the Hahn polynomials rather than the Krawtchouk polynomials. The Hahn polynomials, however, do not have a convenient integral representation and their asymptotics cannot be evaluated by classical means. This obstacle was overcome by Baik et al., [BKMM], who extended the Riemann–Hilbert/steepest descent method 146 Percy Deift in [DKMVZ1] and [DKMVZ2] to a general class of discrete orthogonal polynomials. In this way they were able to compute the asymptotics of the Hahn polynomials and verify F2-behavior for the fluctuations of the temperate zone, as in the case of the Aztec diamond. Problem 7 (Airline boarding). In [BBSSS] the authors show that bN (π), the board- ing time for N passengers subject to the protocol (a)(b)(c) in Problem 7, behaves statistically like the largest eigenvalue of a GUE matrix, √ bN − 2 N lim Prob ≤ t = F (t). (63) N→∞ N 1/6 2 The proof of (63) in [BBSSS] relies on the description of the Robinson–Schensted correspondence in terms of Viennot diagrams (see [Sag]). We illustrate the situation with the permutation π : 341562 in S6 (cf. Problem 3 and (42)). We say that a point (x ,y ) lies in the shadow of a point (x, y) in the plane if x >xand y >y. Plot π as a graph (1, 3), (2, 4),...,(6, 2) in the first quadrant of R2. Consider all the points in the graph which are not in the shadow of any other point: in our case (1, 3) and (3, 1). The first shadow line L1 is the boundary of the combined shadows of these two points (see Figure 3). To form the second shadow line L2, one removes the points (1, 3), (3, 1) on L1, and repeats the procedure, etc. Eventually one obtains kN (π) = k shadow lines L1,...,Lk for some integer k. In our example k = 4, which we note is precisely l6(π), the length of the longest increasing subsequence for π. This is no accident: for any π ∈ SN , we always have kN (π) = lN (π) (see [Sag]). Figure 3. Shadow lines for π : 341562inS6. Universality for mathematical and physical systems 147 The beautiful fact is that each shadow line describes a step in the boarding process. Indeed, noting the y-values of the Lj ’s, we observe that L1 ←→ 3 and 1 are seated L2 ←→ 4 and 2 are seated L3 ←→ 5 is seated L4 ←→ 6 is seated Thus bN (π) = kN (π) = lN (π) = qN (π), and (63) follows from (46). In the language of physics, if we rotate the Viennot diagram for π counterclockwise by 45o, we see that the shadow region of a point on the graph is simply the forward light cone based at that point (speed of light = 1). In other words, for appropriate coordinates a, b we are dealing with the Lorentzian metric ds2 = dadb. In order to incorporate more realistic features into their boarding model, such as the number of seats per row, average amount of aisle length occupied by a passenger, etc., the authors in [BBSSS] observe that it is enough simply to replace ds2 = dadb by a more general Lorentzian metric ds2 = 4D2p(a,b)(dadb+kα(a,b)da2) for appropriate parameters/functions D, p, k and α (see [BBSSS], equation (1)). Thus the basic phenomenon of blocking in the airline boarding problem is modeled in the general case by the forward light cone of some Lorentzian metric. Problems 1 and 2 above, as opposed to 3–7, are purely deterministic and yet it seems that they are well described by a random model, RMT. At first blush, this might seem counterintuitive, but there is a long history of the description of deterministic systems by random models. After all, the throw of a (fair) 6-sided die through the air is completely described by Newton’s laws: Nevertheless, there is no doubt that the right way to describe the outcome is probabilistic, with a one in six chance for each side. With this example in mind, we may say that Wigner was looking for the right stochastic model to describe the neutron scattering “die”. Problems 1–7 above are just a few of the many examples now known of mathe- matical/physical systems which exhibit random matrix type universal behavior. Other systems, from many different areas, can be found for example in [Meh] and the re- views [TraWid3], [For2], and [FerPra]. A particularly fruitful development has been the discovery of connections between random matrix theory and stochastic growth models in the KPZ class ([PraSpo], [FerPra]), and between random matrix theory and equilibrium crystals with short range interactions ([CerKen], [FerSpo], [OkoRes], [FerPraSpo]). In addition, for applications to principal component analysis in statis- tics in situations where the number of variables is comparable to the sample size, see [John] and [BBP] and the references therein. For a relatively recent review of the extensive application of RMT to quantum transport, see [Bee]. Returning to Wigner’s introduction of random matrix theory into theoretical physics, we note that GOE is of course a mathematical model far removed from the laboratory of neutrons colliding with nuclei. Nevertheless, Wigner posited that 148 Percy Deift these two worlds were related: With hindsight, we recognize Wigner’s insight as heralding the emergence of a scientific commonality far across the borders of physics and mathematics. 5. Comments and speculations As is clear from the text, many different kinds of mathematics are needed to analyze Problems 3–7. These include • combinatorial identities, • Riemann–Hilbert methods, • Painlevé theory, • theory of Riemann surfaces, • representation theory, • classical and Riemann–Hilbert steepest descent methods and, most importantly, • random matrix theory. The relevant combinatorial identities are often obtained by analyzing random particle systems conditioned not to intersect, as in Problem 4. The Riemann–Hilbert steepest descent method has its origins in the theory of integrable systems, as in [DeiZho]. There is no space in this article to describe the implementation of any of the above techniques in any detail. Instead, we refer the reader to [Dei2], which is addressed to a general mathematical audience, for a description of the proof of (46) in particular, using Gessel’s formula in combinatorics [Ges], together with the Riemann–Hilbert steepest descent method. For Problem 2 the proofs are based on combinatorial facts and random matrix theory, together with techniques from the theory of L-functions, over Q and also (in [KatSar]) over finite fields. Universality as described in this article poses a challenge to probability theory per se. The central limit theorem (2) above has three components: a statistical compo- nent (take independent, identically distributed random variables, centered and scaled), an algebraic component (add the variables), and an analytic component (take the limit in distribution as n →∞). The outcome of this procedure is then universal – the Gaussian distribution. The challenge to probabilists is to describe an analogous purely probabilistic procedure whose outcome is F1,orF2, etc. The main difficulty is to identify the algebraic component, call it operation X.GivenX, if one takes i.i.d.’s, suitably centered and scaled, performs operation X on them, and then takes the limit in distribution, the outcome should be F1,orF2, etc. Interesting progress has been made recently (see [BodMar] and [BaiSui]) on identifying X for F2. For a different ap- proach to the results in [BodMar] and [BaiSui], see [Sui], where the author uses a very interesting generalized version of the Lindeberg principle due to Chatterjee [Cha1], [Cha2]. Universality for mathematical and physical systems 149 Our final comment/speculation is on the space D, say, of probability distributions. A priori, D is just a set without any “topography”. But we know at least one interesting point on D, the Gaussian distribution FG . 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Our goal is to survey some of the main advances which took place recently in the study of the geometry of projective or compact Kähler manifolds: very efficient new transcendental techniques, a better understanding of the geometric structure of cones of positive cohomology classes and of the deformation theory of Kähler manifolds, new results around the invariance of plurigenera and in the minimal model program. Mathematics Subject Classification (2000). Primary 14C30; Secondary 32C17, 32C30, 32L20. Keywords. Projective variety, Kähler manifold, Hodge theory, positive current, Monge– Ampère equation, Lelong number, Chern connection, curvature, Bochner–Kodaira technique, Kodaira embedding theorem, Kähler cone, ample divisor, nef divisor, pseudo-effective cone, Neron–Severi group, L2 estimates, vanishing theorem, Ohsawa–Takegoshi extension theorem, pluricanonical ring, invariance of plurigenera. 1. Introduction Modern algebraic geometry is one the most intricate crossroads between various branches of mathematics: commutative algebra, complex analysis, global analysis on manifolds, partial differential equations, differential topology, symplectic geometry, number theory…. This interplay has already been strongly emphasized by historical precursors, including Hodge, Kodaira, Hirzebruch and Grauert. Of course, there have been also fruitful efforts to establish purely algebraic foundations of the major results of algebraic geometry, and many prominent mathematicians such as Grothendieck, Deligne and Mumford stand out among the founders of this trend. The present con- tribution stands closer to the above mentioned wider approach; its goal is to explain some recent applications of local and global complex analytic methods to the study of projective algebraic varieties. A unifying theme is the concept of positivity: ample line bundles are characterized by the positivity of their curvature in the complex geometric setting (Kodaira [35]). Projective manifolds thus appear as a subclass of the class of compact Kähler mani- folds, and their cohomological properties can be derived from the study of harmonic forms on Kähler manifolds (Hodge theory). In this vein, another central concept is the concept of positive current, which was introduced by P. Lelong during the 50s. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2007 European Mathematical Society 154 Jean-Pierre Demailly By carefully studying the singularities and the intersection theory of such currents, we derive precise structure theorems for the Kähler cone and for the cone of effective divisors of arbitrary projective varieties ([5], [18]). L2 estimates for solutions of ∂ equations are another crucial technique for proving vanishing theorems for the cohomology of holomorphic vector bundles or sheaves. A combination of the Bochner–Kodaira differential geometric estimate with PDE techniques of Kohn, Hörmander and Andreotti–Vesentini led in the 60s to powerful existence theorems for ∂-equations in hermitian vector bundles of positive curvature. A more recent and equally decisive outcome is the L2 extension theorem by Ohsawa and Takegoshi [48] in 1987. Among applications, we have various forms of approx- imation theorems (closed positive (1, 1)-currents can be approximated by algebraic divisors, and their singularities can be approximated by algebraic singularities). In the analytic setting, this turns out to be the key for the study of adjunction theory (gen- eration properties of adjoint linear systems KX + L, pluricanonical embeddings…). As an illustration, we present a recent proof, adapted from work by Y. T. Siu [58], [59], S. Takayama [62] and M. Pˇaun [52], of the deformation invariance of plurigenera 0 h (Xt ,mKXt ), for an arbitrary projective family (Xt ) of algebraic varieties. 2. Basic concepts and results of complex geometry This section mostly contains only well-known definitions and results. However, we want to fix the notation and describe in detail our starting point. 2.1. Forms, currents, Kähler metrics. Let X be a compact complex manifold and n = X z = (z ,...,z ) dimC . In any local holomorphic coordinate system 1 n , a differ- ential form u of type (p, q) can be written as a sum u(z) = |J |=p,|K|=q uJK(z) dzJ ∧ dzK extended to all increasing multi-indices J, K of length p, q, with the usual no- dz = dz ∧···∧dz tation J j1 jp . We are especially interested in positive currents of type (p, p) p2 T = i TJK(z)dzJ ∧ dzK . |J |=|K|=p Recall that a current is a differential form with distribution coefficients, and that a current is said to be positive if the distribution λj λkTJK is a positive real measure for all complex numbers λJ (which implies TKJ = TJK, hence T = T ). The coefficients TJK are then complex measures – and the diagonal ones TJJ are positive (real) measures. A current is said to be closed if dT = 0 in the sense of distributions. Important examples of closed positive (p, p)-currents are currents of integration over codimen- sion p analytic cycles [A]= cj [Aj ] where the current [Aj ] is defined by duality Kähler manifolds and transcendental techniques in algebraic geometry 155 as [Aj ],u= u|Aj Aj for every (n−p, n−p) test form u on X. Another important example of (1, 1)-current is the Hessian form T = i∂∂ϕ of a plurisubharmonic function on an open set ⊂ X (plurisubharmonic functions are upper semi-continuous functions satisfying the mean value inequality on complex analytic disc; they are characterized by positivity of i ∂2ϕ/∂zj ∂zk dzj ∧ dzk). A Kähler metric on X is a positive definite hermitian (1, 1)-form ω(z) = i ωjk(z)dzj ∧ dzk such that dω = 0, 1≤j,k≤n with smooth coefficients. The manifold X is said to be Kähler if it possesses at least one Kähler metric ω. It is clear that every complex analytic and locally closed N submanifold X ⊂ PC is Kähler (the restriction of the Fubini–Study metric ωFS = i (|z |2 +|z |2 +···+|z |2) X 2π log 0 1 N to is a Kähler metric). Especially projective algebraic varieties are Kähler. 2.2. Cohomology of compact Kähler manifolds. To every d-closed complex val- ued k-form or current α (resp. to every ∂-closed complex valued (p, q)-form or current α) is associated its De Rham (resp. Dolbeault) cohomology class p+q {α}∈H (X, C) (resp. H p,q(X, C)). This definition hides a nontrivial result, namely the fact that all cohomology groups involved (De Rham, Dolbeault, ...) can be defined either in terms of smooth forms or in terms of currents. In fact, if we consider the associated complexes of sheaves, forms and currents both provide acyclic resolutions of the same sheaf (locally constant functions, resp. holomorphic sections). One of the main results of Hodge theory, historically obtained by W. V. D. Hodge through the theory of harmonic forms, is the following fundamental Theorem 2.1. Let (X, ω) be a compact Kähler manifold. Then there is a canonical isomorphism H k(X, C) = H p,q(X, C), p+q=k where each group H p,q(X, C) can be viewed as the space of (p, q)-forms α which are harmonic with respect to ω, i.e. ωα = 0. Now observe that every analytic cycle A = λj Aj of codimension p with integral coefficients defines a cohomology class p,p p p,p p {[A]} ∈ H (X, C) ∩ H 2 (X, Z)/{torsion}⊂H (X, C) ∩ H 2 (X, Q) 156 Jean-Pierre Demailly where H 2p(X, Z)/{torsion}⊂H 2p(X, Q) ⊂ H 2p(X, C) denotes the image of inte- gral classes in complex cohomology. When X is a projective algebraic manifold, this observation leads to the following statement, known as the Hodge conjecture (which was to become one of the famous seven Millenium problems of the Clay Mathematics Institute). Conjecture 2.2. Let X be a projective algebraic manifold. Then the space of “Hodge classes” H p,p(X, C) ∩ H 2p(X, Q) of type (p, p) is generated by classes of algebraic cycles of codimension p with Q-coefficients. At present not much is known to support the positive direction of the Hodge conjecture, not even the case of abelian varieties (i.e. projective algebraic complex tori X = C/ ) – which is the reason why several experts believe that the conjecture could eventually lead to a counterexample. There are however a number of cases where the cohomology algebra can be explicitly computed in terms of the geometry, and which do satisfy the conjecture: flag manifolds (Schubert cycles generate the cohomology ring), moduli spaces of stable or parabolic bundles over a general curve (I. Biswas and M. S. Narasimhan [2]). In the Kähler case the conjecture is trivially wrong as shown by a general complex torus possessing a line bundle with indefinite curvature. Moreover, by a recent result of C.Voisin[66], even a considerably weakened form of the conjecture – adding Chern classes of arbitrary coherent analytic sheaves to the pool of potential generators – is false for non projective complex tori: Theorem 2.3 (C. Voisin [66]). There exists a 4-dimensional complex torus X which possesses a non trivial Hodge class of degree 4, such that every coherent analytic sheaf F on X satisfies c2(F ) = 0. The idea is to show the existence of a 4-dimensional complex torus X = C4/ which does not contain any analytic subset of positive dimension, and such that the Hodge classes of degree 4 are perpendicular to ωn−2 for a suitable choice of the Kähler metric ω. The lattice is explicitly found via a number theoretic construction of Weil based on the number field Q[i], also considered by S. Zucker [70]. The theorem of existence of Hermitian Yang–Mills connections for stable bundles combined with Lübke’s inequality then implies c2(F ) = 0 for every coherent sheaf F on the torus. 2.3. Fundamental L2 existence theorems. Let X be a complex manifold and (E, h) r a hermitian holomorphic vector bundle of rank r over X.IfE|U U × C is a local holomorphic trivialization, the hermitian product can be written as u, v=t uH(z)v where H(z) is the hermitian matric of h and u, v ∈ Ez. It is well known that there exists a unique “Chern connection” D = D1,0 + D0,1 such that D0,1 = ∂ and such that D is compatible with the hermitian metric; in the given trivialization we have , , , − D1 0u = ∂u+ 1 0 ∧u where 1 0 = H 1∂H, and its curvature operator E,h = D2 1,1 ∗ −1 is the smooth section of TX ⊗ Hom(E, E) given by E,h = ∂(H ∂H).IfE is Kähler manifolds and transcendental techniques in algebraic geometry 157 of rank r = 1, then it is customary to write H(z) = e−ϕ(z), and the curvature tensor then takes the simple expression E,h = ∂∂ϕ . In that case the first Chern class of E c (E) = i ∈ H 1,1(X, C) is the cohomology class 1 2π E,h , which is also an integral class in H 2(X, Z)). p,q ∗ In case (X, ω) is a Kähler manifold, the bundles TX ⊗ E are equipped with the hermitian metric induced by p,qω ⊗ h, and we have a Hilbert space of global L2 n sections over X by integrating with respect to the Kähler volume form dVω = ω /n!. If A, B are differential operators acting on L2 space of sections (in general, they are just closed and densely defined operators), we denote by A∗ the formal adjoint of A, and by [A, B]=AB − (−1)deg A deg B BA the usual commutator bracket of operators. The fundamental operator ω of Kähler geometry is the adjoint of the wedge multiplication operator u → ω ∧ u. In this context we have the following fundamental existence theorems for ∂- equations, which is the culmination of several decades of work by Bochner [3], Koda- ira [35], Kohn [37], Andreotti–Vesentini [1], Hörmander [25], Skoda [60], Ohsawa– Takegoshi [48] (and many others). The proofs always proceed through differential geometric inequalities relating the Laplace–Beltrami operators with the curvature (Bochner–Kodaira identities and inequalities). The most basic result is the L2 exis- tence theorem for solutions of ∂-equations. Theorem 2.4 ([1], see also [10]). Let (X, ω) be a Kähler manifold which is “com- plete” in the sense that it possesses a geodesically complete Kähler metric ω. Let E be a hermitian holomorphic vector bundle of rank r over X, and assume that the curva- p,q p,q ture operator AE,h,ω =[i E,h, ω] is positive definite everywhere on TX ⊗ E, 2 p,q q ≥ 1. Then for any form g ∈ L (X, TX ⊗ E) satisfying ∂g = 0 and p,q −1 2 p,q−1 X(AE,h,ω) g,g dVω < +∞, there exists f ∈ L (X, TX ⊗ E) such that ∂f = g and 2 p,q −1 |f | dVω ≤ (AE,h,ω) g,g dVω. X X p,q It is thus of crucial importance to study conditions under which the operator AE,h,ω is positive definite. An easier case is when E is a line bundle. Then we denote by γ1(z) ≤···≤γn(z) the eigenvalues of the real (1, 1)-form i E,h(z) with respect to the metric ω(z) at each point. A straightforward calculation shows that p,q 2 AE,h,ωu, u= γk − γj |uJK| . |J |=p,|K|=q k∈K j∈J In particular, for (n, q)-forms the negative sum − j∈J γj disappears and we have n,q 2 n,q −1 −1 2 AE,h,ωu, u≥(γ1 +···+γq )|u| , (AE,h,ω) u, u≤(γ1 +···+γq ) |u| provided the line bundle (E, h) has positive definite curvature. Therefore ∂-equations can be solved for all L2 (n, q)-forms with q ≥ 1, and this is the major reason 158 Jean-Pierre Demailly why vanishing results for H q cohomology groups are usually obtained for sections n ∗ n of the “adjoint line bundle” E = KX ⊗ E, where KX = TX = X is the “canonical bundle”ofX, rather than for E itself. Especially, if X is compact (or q weakly pseudoconvex) and i E,h > 0, then H (X, KX⊗E) = 0 for q ≥ 1 (Kodaira), p,q and more generally H (X, E) = 0 for p + q ≥ n + 1 (Kodaira–Nakano, take ω = i E,h, in which case γj ≡ 1 for all j and k∈K γk − j∈J γj = p + q − n). As shown in [10], Theorem 2.4 still holds true in that case when h is a singular hermitian metric, i.e. a metric whose weights ϕ are arbitrary locally integrable func- tions, provided that the curvature is (E, h) is positive in the sense of currents (i.e., the weights ϕ are strictly plurisubharmonic). This implies the well-known Nadel vanish- ing theorem ([42], [12], [15]), a generalization of the Kawamata–Viehweg vanishing theorem [28], [65]. Theorem 2.5 (Nadel). Let (X, ω) be a compact (or weakly pseudoconvex) Kähler manifold, and (L, h) a singular hermitian line bundle such that L,h ≥ εω for some q ε>0. Then H (X, KX ⊗ L ⊗ (h)) = 0 for q ≥ 1, where (h) is the multiplier ideal sheaf of h, namely the sheaf of germs of holomorphic functions f on X such that |f |2e−ϕ is locally integrable with respect to the local weights h = e−ϕ. It is well known that Theorems 2.4 and 2.5, more specifically its “singular hermi- tian” version, imply almost all other fundamental vanishing or existence theorems of algebraic geometry, as well as their analytic counterparts in the framework of Stein manifolds (general solution of the Levi problem by Grauert), see e.g. Demailly [16] for a recent account. In particular, one gets as a consequence the Kodaira embedding theorem [35]. Theorem 2.6. Let X be a compact complex n-dimensional manifold. Then the fol- lowing properties are equivalent. N (i) X can be embedded in some projective space PC as a closed analytic submani- fold (and such a submanifold is automatically algebraic by Chow’s theorem). (ii) X carries a hermitian holomorphic line bundle (L, h) with positive definite smooth curvature form i L,h > 0. (iii) X possesses a Hodge metric, i.e., a Kähler metric ω such that {ω}∈H 2(X, Z). ⊗m If property (ii) holds true, then for m ≥ m0 1 the multiple L is very ample, namely we have an embedding given by the linear system V = H 0(X, L⊗m) of sections, ∗ L⊗m : X −→ P(V ), z → Hz ={σ ∈ V ; σ(z) = 0}⊂V, L⊗m ∗ O( ) P(V∗) and L⊗m 1 is the pull-back of the canonical bundle on . Another fundamental existence theorem is the L2-extension result by Ohsawa– Takegoshi [48]. Many different versions and generalizations have been given in recent years [43], [44], [45], [46], [47]. Here is another one, due to Manivel [40], which is slightly less general but simpler to state. Kähler manifolds and transcendental techniques in algebraic geometry 159 Theorem 2.7 (Ohsawa–Takegoshi [48], Manivel [40]). Let X be a compact or weakly pseudoconvex n-dimensional complex manifold equipped with a Kähler metric ω, let L (resp. E) be a hermitian holomorphic line bundle (resp. a hermitian holomorphic vector bundle of rank r over X), and s a global holomorphic section of E. Assume that s is generically transverse to the zero section, and let r Y = x ∈ X ; s(x) = 0, ds(x) = 0 ,p= dim Y = n − r. Moreover, assume that the (1, 1)-form i (L)+ri∂∂ log |s|2 is semipositive and that there is a continuous function α ≥ 1 such that the following two inequalities hold everywhere on X: − {i (E)s, s} (i) i (L) + ri∂∂ log |s|2 ≥ α 1 , |s|2 (ii) |s|≤e−α. Then for every holomorphic section f over Y of the adjoint line bundle L = KX ⊗ L 2 r −2 (restricted to Y), such that Y |f | | (ds)| dVω < +∞, there exists a holomorphic extension F of f over X, with values in L , such that |F |2 |f |2 dV ≤ C dV , r X,ω r r Y,ω X |s|2 (− log |s|)2 Y | (ds)|2 where Cr is a numerical constant depending only on r. The proof actually shows that the extension theorem holds true as well for ∂-closed (0,q)-forms with values in L , of which the stated theorem is the special case q = 0. There are several other important L2 existence theorems. One of them is Skoda’s criterion for the surjectivity of holomorphic bundle morphisms – more concretely, a Bezout type division theorem for holomorphic function. It can be derived either from Theorem 2.4 on ∂-equations through sharp curvature calculations (this is Skoda’s original approach in [60]), or as a consequence of the above extension theorem 2.7 (see Ohsawa [46]). 2.4. Positive cones. We now introduce some further basic objects of projective or Kähler geometry, namely cones of positive cohomology classes. Definition 2.8. Let X be a compact Kähler manifold and H 1,1(X, R) the space of real (1, 1) cohomology classes. (i) The Kähler cone is the set K ⊂ H 1,1(X, R) of cohomology classes {ω} of Kähler forms. This is clearly an open convex cone. (ii) The pseudo-effective cone is the set E ⊂ H 1,1(X, R) of cohomology classes {T } of closed positive currents of type (1, 1). This is a closed convex cone (as follows from the weak compactness property of bounded sets of positive measures or currents). 160 Jean-Pierre Demailly It follows from this definition that K ⊂ E. In general the inclusion is strict. p To see this, it is enough to observe that a Kähler class {α} satisfies Y α > 0 for every p-dimensional analytic set. On the other hand, if X is the surface obtained by 2 E P1 blowing-up P in one point, then the exceptional divisor has a cohomology 2 class {α} such that E α = E =−1, hence {α} ∈/ K, although {α}={[E]} ∈ E. In case X is projective it is interesting to consider also the algebraic analogues of our “transcendental cones” K and E, which consist of suitable integral divisor classes. Since the cohomology classes of such divisors live in H 2(X, Z), we are led to introduce the Neron–Severi lattice and the associated Neron–Severi space: , NS(X) := H 1 1(X, R) ∩ H 2(X, Z)/{torsion} , NSR(X) := NS(X) ⊗Z R. All classes of real divisors D = cj Dj , cj ∈ R, lie by definition in NSR(X). Notice that the integral lattice H 2(X, Z)/{torsion} need not hit at all the subspace H 1,1(X, R) ⊂ H 2(X, R) in the Hodge decomposition, hence in general the Picard number, defined as ρ(X) = rankZ NS(X) = dimR NSR(X), , , satisfies ρ(X) ≤ h1 1 = dimR H 1 1(X, R), but the equality can be strict (actually, it is well known that a generic complex torus X = Cn/ satisfies ρ(X) = 0 and h1,1 = n2). In order to deal with the case of algebraic varieties we introduce KNS = K ∩ NSR(X), ENS = E ∩ NSR(X). A very important fact is that the “Neron–Severi part” of any of the open or closed transcendental cones K, E, K, E is algebraic, i.e. can be characterized in simple algebraic terms. Theorem 2.9. Let X be a projective manifold. Then E (i) NS is the closure of the cone generated by classes of effective divisors, i.e. divisors D = cj Dj , cj ∈ R+. (ii) KNS is the open cone generated by classes of ample (or very ample) divisors A (recall that a divisor A is said to be very ample if the linear system H 0(X, O(A)) provides an embedding of X in projective space). E (iii) The interior NS is the cone generated by classes of big divisors, namely divisors D such that h0(X, O(kD)) ≥ ckdim X for k large. (iv) The closed cone KNS consists of the closure of the cone generated by nef divisors D(or nef line bundles L), namely effective integral divisors D such that D · C ≥ 0 for every curve C. Kähler manifolds and transcendental techniques in algebraic geometry 161 By extension, we will say that K is the cone of nef (1, 1)-cohomology classes (even though they are not necessarily integral). Sketch of proof (see also [13] for more details). If we denote by Kalg the open cone generated by ample divisors, resp. by Ealg the closure of the cone generated by effective divisors, we have Kalg ⊂ KNS, Ealg ⊂ ENS, and clearly the interesting part lies in the converse inclusions. The inclusion KNS ⊂ Kalg is equivalent to the Kodaira embedding theorem: if a rational class {α} is in K, then some multiple of {α} is the first Chern class of a hermitian line bundle L whose curvature form is Kähler. Therefore L is ample and {α}∈Kalg; property (ii) follows. {α}∈E {α − εω}∈E Similarly, if we take a rational class NS, then we still have NS by subtracting a small multiple εω of a Kähler class, hence α − εω ≡ T ≥ 0 for some positive current T . Therefore some multiple {m0α} is the first Chern class of a hermitian line bundle (L, h) with curvature current T : i L,h := − i∂∂ log h = m (T + εω) ≥ m εω. 2π 0 0 Theorem 2.4 on L2 estimates for ∂-equations then shows that large multiples L⊗k admit a large number of sections, hence L⊗k can be represented by a big divisor. E ⊂ E E ⊂ E This implies (iii) and also that NS alg. Therefore NS alg by passing to the closure; (i) follows. The statement (iv) about nef divisors follows e.g. from Klaiman [34] and Hartshorne [24], since every nef divisor is a limit of a sequence of ample rational divisors. 2 As a natural extrapolation of the algebraic situation, we say that K is the cone of nef (1, 1)-cohomology classes (even though these classes are not necessarily integral). Property 2.9 (i) also explains the terminology used for the pseudo-effective cone. 2.5. Approximation of currents and Zariski decomposition. Let X be compact Kähler manifold and let α ∈ E be in the interior of the pseudo-effective cone. In analogy with the algebraic context, such a class α is called “big”, and it can then be represented by a Kähler current T , i.e. a closed positive (1, 1)-current T such that T ≥ δω for some smooth hermitian metric ω and a constant δ 1. Notice that the latter definition of a Kähler current makes sense even if X is an arbitrary (non necessarily Kähler) compact complex manifold. Theorem 2.10 (Demailly [14], Boucksom [4], 3.1.24). If T is a Kähler current on a compact complex manifold X, then one can write T = lim Tm for a sequence of Kähler currents Tm in the same cohomology class as T , which have logarithmic poles 1 and coefficients in m Z. This means that there are modifications μm : Xm → X such that μmTm =[Em]+βm 1 where Em is an effective Q-divisor on Xm with coefficients in m Z (Em is the “fixed part” and βm a closed semi-positive form, the “movable part”). 162 Jean-Pierre Demailly Proof. We just recall the main idea and refer to [14] for details. Locally we can write T = i∂∂ϕ for some strictly plurisubharmonic potential ϕ on X. The approximating potentials ϕm of ϕ are defined as ϕ (z) = 1 |g (z)|2 m m log ,m 2 where (g,m) is a Hilbert basis of the space H(, mϕ) of holomorphic functions which are L2 with respect to the weight e−2mϕ. The Ohsawa–Takegoshi L2 extension theorem 2.7 (applied to extension from a single isolated point) implies that there are enough such holomorphic functions, and thus ϕm ≥ ϕ − C/m. On the other hand ϕ = limm→+∞ ϕm by a Bergman kernel trick and by the mean value inequality. The Hilbert basis (g,m) is also a family of local generators of the globally defined multiplier ideal sheaf (mT ) = (mϕ). The modification μm : Xm → X is obtained by blowing-up this ideal sheaf, so that μm (mT ) = O(−mEm) for some effective Q-divisor Em with normal crossings on Xm. Now we set Tm = ∗ i∂∂ϕm and βm = μmTm −[Em]. Then βm = i∂∂ψm where ψ = 1 |g μ /h|2 X m m log ,m m locally on m 2 and h is a generator of O(−mEm), and we see that βm is a smooth semi-positive form on Xm. The construction can be made global by using a gluing technique, e.g. via partitions of unity. 2 Remark 2.11. The more familiar algebraic analogue would be to take α = c1(L) with a big line bundle L and to blow-up the base locus of |mL|, m 1, to get a Q-divisor decomposition μmL ∼ Em + Dm,Em effective,Dm free. Such a blow-up is usually referred to as a “log resolution” of the linear system |mL|, and we say that Em + Dm is an approximate Zariski decomposition of L. We will also use this terminology for Kähler currents with logarithmic poles. In the above construction βm is not just semi-positive, it is even positive definite on tangent vectors which are not mapped to 0 by the differential dμm, in particular βm is positive definite outside the exceptional divisor. However, if E is the exceptional divisor of the blow-up along a smooth centre Y ⊂ X, then O(−E) is relatively ample with respect to the blow-up map π, hence the negative current −[E] is cohomologous to a smooth form θE which is positive along the fibers of π. As a consequence, we can slightly perturb the decomposition of α by increasing multiplicities in the Kähler manifolds and transcendental techniques in algebraic geometry 163